2. Preliminaries

Let $\mathbb{Z}_+$ , $\mathbb{N}$ , $\mathbb{R}$ , $\mathbb{R}_+$ , $\mathbb{R}_{++}$ , and $\mathbb{C}$ denote the sets of non-negative integers, positive integers, real numbers, non-negative real numbers, positive real numbers, and complex numbers, respectively. For $x , y \in \mathbb{R}$ , we will use the notation $x \land y \,:\!=\, \min \{x, y\}$ , $x \lor y \,:\!=\, \max \{x, y\}$ , and $x^+ \,:\!=\, \max \{0, x\}$ . By $\langle{{\boldsymbol{{x}}}}, {{\boldsymbol{{y}}}}\rangle \,:\!=\, \sum_{j=1}^d x_j \overline{y_j}$ we denote the Euclidean inner product of ${{\boldsymbol{{x}}}} = (x_1, \ldots, x_d)^\top \in \mathbb{C}^d$ and ${{\boldsymbol{{y}}}} = (y_1, \ldots, y_d)^\top \in \mathbb{C}^d$ , and by $\|{{\boldsymbol{{x}}}}\|$ and $\|{\boldsymbol{{A}}}\|$ we denote the induced norms of ${{\boldsymbol{{x}}}} \in \mathbb{C}^d$ and ${\boldsymbol{{A}}} \in \mathbb{C}^{d\times d}$ , respectively. By $r({\boldsymbol{{A}}})$ we denote the spectral radius of ${\boldsymbol{{A}}} \in \mathbb{C}^{d\times d}$ . The null vector and the null matrix will be denoted by ${\boldsymbol{0}}$ . Moreover, ${{\boldsymbol{{I}}}}_d \in \mathbb{R}^{d\times d}$ denotes the identity matrix. If ${\boldsymbol{{A}}} \in \mathbb{R}^{d\times d}$ is positive semidefinite, then ${\boldsymbol{{A}}}^{1/2}$ denotes the unique positive semidefinite square root of ${\boldsymbol{{A}}}$ . If ${\boldsymbol{{A}}} \in \mathbb{R}^{d\times d}$ is strictly positive definite, then ${\boldsymbol{{A}}}^{1/2}$ is strictly positive definite and ${\boldsymbol{{A}}}^{-1/2}$ denotes the inverse of ${\boldsymbol{{A}}}^{1/2}$ . The set of $d \times d$ matrices with non-negative off-diagonal entries (also called essentially non-negative matrices) is denoted by $\mathbb{R}^{d\times d}_{(+)}$ . By $C^2_\mathrm{c}(\mathbb{R}_+^d,\mathbb{R})$ we denote the set of twice continuously differentiable real-valued functions on $\mathbb{R}_+^d$ with compact support. By $B(\mathbb{R}_+^d,\mathbb{R})$ we denote the Banach space (endowed with the supremum norm) of real-valued bounded Borel functions on $\mathbb{R}_+^d$ . Convergence almost surely, in $L_1$ , in $L_2$ , in probability, and in distribution will be denoted by $\stackrel{{\mathrm{a.s.}}}{\longrightarrow}$ , $\stackrel{L_1}{\longrightarrow}$ , $\stackrel{L_2}{\longrightarrow}$ , $\stackrel{\mathbb{P}}{\longrightarrow}$ and $\stackrel{{\mathcal D}}{\longrightarrow}$ , respectively. For an event A with $\mathbb{P}(A) > 0$ , let ${\mathbb{P}}_A(\!\cdot\!) \,:\!=\, {\mathbb{P}}(\!\cdot\,|\, A) = {\mathbb{P}}(\!\cdot \cap A) / {\mathbb{P}}(A)$ denote the conditional probability measure given A, and let $\stackrel{{\mathcal D}_A}{\longrightarrow}$ denote convergence in distribution under the conditional probability measure ${\mathbb{P}}_A$ . Almost sure equality and equality in distribution will be denoted by $\stackrel{{\mathrm{a.s.}}}{=}$ and $\stackrel{{\mathcal D}}{=}$ , respectively. If ${{\boldsymbol{{V}}}} \in \mathbb{R}^{d\times d}$ is symmetric and positive semidefinite, then ${\mathcal N}_d({\boldsymbol{0}}, {{\boldsymbol{{V}}}})$ denotes the d-dimensional normal distribution with zero mean and variance matrix ${{\boldsymbol{{V}}}}$ . Throughout this paper, we make the conventions $\int_a^b \,:\!=\, \int_{(a,b]}$ and $\int_a^\infty \,:\!=\, \int_{(a,\infty)}$ for any $a, b \in \mathbb{R}$ with $a < b$ .

1. (i) $d \in \mathbb{N}$ ,

2. (ii) ${{\boldsymbol{{c}}}} = (c_i)_{i\in\{1,\ldots,d\}} \in \mathbb{R}_+^d$ ,

3. (iii) ${\boldsymbol{\beta}} = (\beta_i)_{i\in\{1,\ldots,d\}} \in \mathbb{R}_+^d$ ,

4. (iv) ${{\boldsymbol{{B}}}} = (b_{i,j})_{i,j\in\{1,\ldots,d\}} \in \mathbb{R}^{d \times d}_{(+)}$ ,

5. (v) $\nu$ is a Borel measure on ${\mathcal U}_d \,:\!=\, \mathbb{R}_+^d \setminus \{{\boldsymbol{0}}\}$ satisfying $\int_{{\mathcal U}_d} (1 \land \|{{\boldsymbol{{r}}}}\|) \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) < \infty$ , and

6. (vi) $\boldsymbol{\mu} = (\mu_1, \ldots, \mu_d)$ , where, for each $i \in \{1, \ldots, d\}$ , $\mu_i$ is a Borel measure on ${\mathcal U}_d$ satisfying

   \begin{equation*} \int_{{\mathcal U}_d} \biggl[\|{{\boldsymbol{{z}}}}\| \land \|{{\boldsymbol{{z}}}}\|^2 + \sum_{j \in \{1, \ldots, d\} \setminus \{i\}} (1 \land z_j)\biggr] \mu_i(\mathrm{d}{{\boldsymbol{{z}}}}) < \infty . \end{equation*}

Theorem 2.1. Let $(d, {{\boldsymbol{{c}}}}, {\boldsymbol{\beta}}, {{\boldsymbol{{B}}}}, \nu, \boldsymbol{\mu})$ be a set of admissible parameters. Then there exists a unique conservative transition semigroup $(P_t)_{t\in\mathbb{R}_+}$ acting on $B(\mathbb{R}_+^d,\mathbb{R})$ such that its Laplace transform has a representation

\begin{equation*} \int_{\mathbb{R}_+^d} \mathrm{e}^{- \langle \boldsymbol{\lambda}, {{\boldsymbol{{y}}}} \rangle} P_t({{\boldsymbol{{x}}}}, \mathrm{d} {{\boldsymbol{{y}}}}) = \mathrm{e}^{- \langle {{\boldsymbol{{x}}}}, {{\boldsymbol{{v}}}}(t, \boldsymbol{\lambda}) \rangle - \int_0^t \psi({{\boldsymbol{{v}}}}(s, \boldsymbol{\lambda})) \, \mathrm{d} s} , \qquad {{\boldsymbol{{x}}}} \in \mathbb{R}_+^d, \quad \boldsymbol{\lambda} \in \mathbb{R}_+^d , \quad t \in \mathbb{R}_+ , \end{equation*}

where, for any $\boldsymbol{\lambda} \in \mathbb{R}_+^d$ , the continuously differentiable function $\mathbb{R}_+ \ni t \mapsto {{\boldsymbol{{v}}}}(t, \boldsymbol{\lambda}) = (v_1(t, \boldsymbol{\lambda}), \ldots, v_d(t, \boldsymbol{\lambda}))^\top \in \mathbb{R}_+^d$ is the unique locally bounded solution to the system of differential equations

\begin{equation*} \partial_t v_i(t, \boldsymbol{\lambda}) = - \varphi_i({{\boldsymbol{{v}}}}(t, \boldsymbol{\lambda})) , \qquad v_i(0, \boldsymbol{\lambda}) = \lambda_i , \qquad i \in \{1, \ldots, d\} , \end{equation*}

with

\begin{equation*} \varphi_i(\boldsymbol{\lambda}) \,:\!=\, c_i \lambda_i^2 - \langle {{\boldsymbol{{B}}}} {{\boldsymbol{{e}}}}_i, \boldsymbol{\lambda} \rangle + \int_{{\mathcal U}_d} \bigl( \mathrm{e}^{- \langle \boldsymbol{\lambda}, {{\boldsymbol{{z}}}} \rangle} - 1 + \lambda_i (1 \land z_i) \bigr) \, \mu_i(\mathrm{d} {{\boldsymbol{{z}}}}) \end{equation*}

for $\boldsymbol{\lambda} \in \mathbb{R}_+^d$ , $i \in \{1, \ldots, d\}$ , and

\begin{equation*} \psi(\boldsymbol{\lambda}) \,:\!=\, \langle \boldsymbol{\beta}, \boldsymbol{\lambda} \rangle + \int_{{\mathcal U}_d} \bigl( 1 - \mathrm{e}^{- \langle\boldsymbol{\lambda}, {{\boldsymbol{{r}}}}\rangle} \bigr) \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) , \qquad \boldsymbol{\lambda} \in \mathbb{R}_+^d . \end{equation*}

Theorem 2.1 is a special case of Theorem 2.7 of Duffie et al. [Reference Duffie, Filipović and Schachermayer13] with $m = d$ , $n = 0$ , and zero killing rate. For more details, see Remark 2.5 in Barczy et al. [Reference Barczy, Li and Pap6].

Let $({{\boldsymbol{{X}}}}_t)_{t\in\mathbb{R}_+}$ be a multi-type CBI process with parameters $(d, {{\boldsymbol{{c}}}}, {\boldsymbol{\beta}}, {{\boldsymbol{{B}}}}, \nu, \boldsymbol{\mu})$ such that ${\mathbb{E}}(\|{{\boldsymbol{{X}}}}_0\|) < \infty$ and the moment condition

(2.1) \begin{equation} \int_{{\mathcal U}_d} \|{{\boldsymbol{{r}}}}\| {\mathbb{1}}_{\{\|{{\boldsymbol{{r}}}}\|\geqslant 1\}} \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) < \infty \end{equation}

holds. Then, by the formula (3.4) in Barczy et al. [Reference Barczy, Li and Pap6],

(2.2) \begin{equation} {\mathbb{E}}({{\boldsymbol{{X}}}}_t \,|\, {{\boldsymbol{{X}}}}_0 = {{\boldsymbol{{x}}}}) = \mathrm{e}^{t{\widetilde{{{\boldsymbol{{B}}}}}}} {{\boldsymbol{{x}}}} + \int_0^t \mathrm{e}^{u{\widetilde{{{\boldsymbol{{B}}}}}}} \widetilde{{\boldsymbol{\beta}}} \, \mathrm{d} u , \qquad {{\boldsymbol{{x}}}} \in \mathbb{R}_+^d , \quad t \in \mathbb{R}_+ , \end{equation}

where

\begin{gather*} {\widetilde{{{\boldsymbol{{B}}}}}} \,:\!=\, (\widetilde{b}_{i,j})_{i,j\in\{1,\ldots,d\}} , \quad \widetilde{b}_{i,j} \,:\!=\, b_{i,j} + \int_{{\mathcal U}_d} (z_i - \delta_{i,j})^+ \, \mu_j(\mathrm{d}{{\boldsymbol{{z}}}}) , \quad \widetilde{{\boldsymbol{\beta}}} \,:\!=\, {\boldsymbol{\beta}} + \int_{{\mathcal U}_d} {{\boldsymbol{{r}}}} \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) , \end{gather*}

with $\delta_{i,j}\,:\!=\,1$ if $i = j$ , and $\delta_{i,j} \,:\!=\, 0$ if $i \ne j$ . Note that, for each ${{\boldsymbol{{x}}}} \in \mathbb{R}^d_+$ , the function $\mathbb{R}_+ \ni t \mapsto {\mathbb{E}}({{\boldsymbol{{X}}}}_t \,|\, {{\boldsymbol{{X}}}}_0 = {{\boldsymbol{{x}}}})$ is continuous, and ${\widetilde{{{\boldsymbol{{B}}}}}} \in \mathbb{R}^{d \times d}_{(+)}$ and $\widetilde{{\boldsymbol{\beta}}} \in \mathbb{R}_+^d$ , since

\begin{equation*} \int_{{\mathcal U}_d} \|{{\boldsymbol{{r}}}}\| \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) < \infty , \qquad \int_{{\mathcal U}_d} (z_i - \delta_{i,j})^+ \, \mu_j(\mathrm{d} {{\boldsymbol{{z}}}}) < \infty , \quad i, j \in \{1, \ldots, d\}; \end{equation*}

see Barczy et al. [Reference Barczy, Li and Pap6, Section 2]. Further, ${\mathbb{E}}({{\boldsymbol{{X}}}}_t \,|\, {{\boldsymbol{{X}}}}_0 = {{\boldsymbol{{x}}}})$ , ${{\boldsymbol{{x}}}} \in \mathbb{R}_+^d$ , does not depend on the parameter ${{\boldsymbol{{c}}}}$ . One can give probabilistic interpretations of the modified parameters ${\widetilde{{{\boldsymbol{{B}}}}}}$ and $\widetilde{{\boldsymbol{\beta}}}$ : namely, for each $t \in \mathbb{R}_+$ , we have $\mathrm{e}^{t{\widetilde{{{\boldsymbol{{B}}}}}}} {{\boldsymbol{{e}}}}_j = {\mathbb{E}}({{\boldsymbol{{Y}}}}_t \,|\, {{\boldsymbol{{Y}}}}_0 = {{\boldsymbol{{e}}}}_j)$ , $j \in \{1, \ldots, d\}$ , and $t \widetilde{{\boldsymbol{\beta}}} = {\mathbb{E}}({{\boldsymbol{{Z}}}}_t \,|\, {{\boldsymbol{{Z}}}}_0 = {\boldsymbol{0}})$ , where $({{\boldsymbol{{Y}}}}_t)_{t\in\mathbb{R}_+}$ and $({{\boldsymbol{{Z}}}}_t)_{t\in\mathbb{R}_+}$ are multi-type CBI processes with parameters $(d, {{\boldsymbol{{c}}}}, {\boldsymbol{0}}, {{\boldsymbol{{B}}}}, 0, \boldsymbol{\mu})$ and $(d, {\boldsymbol{0}}, {\boldsymbol{\beta}}, {\boldsymbol{0}}, \nu, {\boldsymbol{0}})$ , respectively; see the formula (2.2). The processes $({{\boldsymbol{{Y}}}}_t)_{t\in\mathbb{R}_+}$ and $({{\boldsymbol{{Z}}}}_t)_{t\in\mathbb{R}_+}$ can be viewed as pure branching (without immigration) and pure immigration (without branching) processes, respectively. Consequently, $\mathrm{e}^{\widetilde{{{\boldsymbol{{B}}}}}}$ and $\widetilde{{\boldsymbol{\beta}}}$ may be called the branching mean matrix and the immigration mean vector, respectively. Note that the branching mechanism depends only on the parameters ${{\boldsymbol{{c}}}}$ , ${{\boldsymbol{{B}}}}$ , and $\boldsymbol{\mu}$ , while the immigration mechanism depends only on the parameters ${\boldsymbol{\beta}}$ and $\nu$ .

If $(d, {{\boldsymbol{{c}}}}, {\boldsymbol{\beta}}, {{\boldsymbol{{B}}}}, \nu, \boldsymbol{\mu})$ is a set of admissible parameters, ${\mathbb{E}}(\|{{\boldsymbol{{X}}}}_0\|) < \infty$ , and the moment condition (2.1) holds, then the multi-type CBI process having parameters $(d, {{\boldsymbol{{c}}}}, {\boldsymbol{\beta}}, {{\boldsymbol{{B}}}}, \nu, \boldsymbol{\mu})$ can be represented as a pathwise unique strong solution of the stochastic differential equation

(2.3) \begin{align} \begin{split} {{\boldsymbol{{X}}}}_t &={{\boldsymbol{{X}}}}_0 + \int_0^t ({\boldsymbol{\beta}} + {\widetilde{{{\boldsymbol{{B}}}}}} {{\boldsymbol{{X}}}}_u) \, \mathrm{d} u + \sum_{\ell=1}^d \int_0^t \sqrt{2 c_\ell \max \{0, X_{u,\ell}\}} \, \mathrm{d} W_{u,\ell} \, {{\boldsymbol{{e}}}}_\ell \\ &\quad + \sum_{\ell=1}^d \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} {{\boldsymbol{{z}}}} {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, \widetilde{N}_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) + \int_0^t \int_{{\mathcal U}_d} {{\boldsymbol{{r}}}} \, M(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{r}}}}) \end{split} \end{align}

for $t \in\mathbb{R}_+$ . (See Theorem 4.6 and Section 5 in Barczy et al. [Reference Barczy, Li and Pap6]; there, (2.3) was proved only for $d \in \{1, 2\}$ , but their method clearly works for all $d \in \mathbb{N}$ .) Here $X_{t,\ell}$ , $\ell\in\{1,\ldots,d\}$ , denotes the $\ell$ th coordinate of ${{\boldsymbol{{X}}}}_t$ , ${\mathbb{P}}({{\boldsymbol{{X}}}}_0\in\mathbb{R}_+^d)=1$ ; $(W_{t,1})_{t\in\mathbb{R}_+}$ , …, $(W_{t,d})_{t\in\mathbb{R}_+}$ are standard Wiener processes; and $N_\ell$ , $\ell \in \{1, \ldots, d\}$ , and M are Poisson random measures on $\mathbb{R}_{++} \times {\mathcal U}_d \times \mathbb{R}_{++}$ and on $\mathbb{R}_{++} \times {\mathcal U}_d$ with intensity measures $\mathrm{d} u \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \, \mathrm{d} w$ , $\ell \in \{1, \ldots, d\}$ , and $\mathrm{d} u \, \nu(\mathrm{d}{{\boldsymbol{{r}}}})$ , respectively, such that ${{\boldsymbol{{X}}}}_0$ , $(W_{t,1})_{t\in\mathbb{R}_+}$ , …, $(W_{t,d})_{t\in\mathbb{R}_+}$ , $N_1,\dots,N_d$ , and M are independent, and $\widetilde{N}_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) \,:\!=\, N_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) - \mathrm{d} u \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \, \mathrm{d} w$ , $\ell \in \{1, \ldots, d\}$ .

Next we recall a classification of multi-type CBI processes. For a matrix ${\boldsymbol{{A}}} \in \mathbb{R}^{d \times d}$ , $\sigma({\boldsymbol{{A}}})$ will denote the spectrum of ${\boldsymbol{{A}}}$ , that is, the set of all $\lambda \in \mathbb{C}$ that are eigenvalues of ${\boldsymbol{{A}}}$ . Then $r({\boldsymbol{{A}}}) = \max_{\lambda \in \sigma({\boldsymbol{{A}}})} |\lambda|$ is the spectral radius of ${\boldsymbol{{A}}}$ . Moreover, we will use the notation

\begin{equation*} s({\boldsymbol{{A}}}) \,:\!=\, \max_{\lambda \in \sigma({\boldsymbol{{A}}})} {\operatorname{Re}}(\lambda) . \end{equation*}

A matrix ${\boldsymbol{{A}}} \in \mathbb{R}^{d\times d}$ is called reducible if there exist a permutation matrix ${{\boldsymbol{{P}}}} \in \mathbb{R}^{ d \times d}$ and an integer r with $1 \leqslant r \leqslant d-1$ such that

\begin{equation*} {{\boldsymbol{{P}}}}^\top {\boldsymbol{{A}}} {{\boldsymbol{{P}}}} = \begin{pmatrix} {\boldsymbol{{A}}}_1\;\;\;\;\; & {\boldsymbol{{A}}}_2 \\[5pt] {\boldsymbol{0}}\;\;\;\;\; & {\boldsymbol{{A}}}_3 \end{pmatrix}, \end{equation*}

where ${\boldsymbol{{A}}}_1 \in \mathbb{R}^{r\times r}$ , ${\boldsymbol{{A}}}_3 \in \mathbb{R}^{ (d-r) \times (d-r) }$ , ${\boldsymbol{{A}}}_2 \in \mathbb{R}^{r \times (d-r) }$ , and ${\boldsymbol{0}} \in \mathbb{R}^{(d-r)\times r}$ is a null matrix. A matrix ${\boldsymbol{{A}}} \in \mathbb{R}^{d\times d}$ is called irreducible if it is not reducible; see, e.g., Horn and Johnson [Reference Horn and Johnson15, Definitions 6.2.21 and 6.2.22]. We do emphasize that no 1-by-1 matrix is reducible.

Recall that if ${\widetilde{{{\boldsymbol{{B}}}}}} \in \mathbb{R}^{d\times d}_{(+)}$ is irreducible, then $\mathrm{e}^{t{\widetilde{{{\boldsymbol{{B}}}}}}} \in \mathbb{R}^{d \times d}_{++}$ for all $t \in \mathbb{R}_{++}$ , and $s({\widetilde{{{\boldsymbol{{B}}}}}})$ is a real eigenvalue of ${\widetilde{{{\boldsymbol{{B}}}}}}$ , the algebraic and geometric multiplicities of $s({\widetilde{{{\boldsymbol{{B}}}}}})$ are 1, and the real parts of the other eigenvalues of ${\widetilde{{{\boldsymbol{{B}}}}}}$ are less than $s({\widetilde{{{\boldsymbol{{B}}}}}})$ . Moreover, corresponding to the eigenvalue $s({\widetilde{{{\boldsymbol{{B}}}}}})$ there exists a unique (right) eigenvector ${\widetilde{{{\boldsymbol{{u}}}}}} \in \mathbb{R}^d_{++}$ of ${\widetilde{{{\boldsymbol{{B}}}}}}$ such that the sum of its coordinates is 1; this is also the unique (right) eigenvector of $\mathrm{e}^{\widetilde{{{\boldsymbol{{B}}}}}}$ , called the right Perron vector of $\mathrm{e}^{\widetilde{{{\boldsymbol{{B}}}}}}$ , corresponding to the eigenvalue $r(\mathrm{e}^{\widetilde{{{\boldsymbol{{B}}}}}}) = \mathrm{e}^{s({\widetilde{{{\boldsymbol{{B}}}}}})}$ of $\mathrm{e}^{\widetilde{{{\boldsymbol{{B}}}}}}$ such that the sum of its coordinates is 1. Further, there exists a unique left eigenvector ${{\boldsymbol{{u}}}} \in \mathbb{R}^d_{++}$ of ${\widetilde{{{\boldsymbol{{B}}}}}}$ corresponding to the eigenvalue $s({\widetilde{{{\boldsymbol{{B}}}}}})$ with ${\widetilde{{{\boldsymbol{{u}}}}}}^\top {{\boldsymbol{{u}}}} = 1$ , which is also the unique (left) eigenvector of $\mathrm{e}^{\widetilde{{{\boldsymbol{{B}}}}}}$ , called the left Perron vector of $\mathrm{e}^{\widetilde{{{\boldsymbol{{B}}}}}}$ , corresponding to the eigenvalue $r(\mathrm{e}^{\widetilde{{{\boldsymbol{{B}}}}}}) = \mathrm{e}^{s({\widetilde{{{\boldsymbol{{B}}}}}})}$ of $\mathrm{e}^{\widetilde{{{\boldsymbol{{B}}}}}}$ such that ${\widetilde{{{\boldsymbol{{u}}}}}}^\top {{\boldsymbol{{u}}}} = 1$ . Moreover, there exist $C_1, C_2, C_3, C_4 \in \mathbb{R}_{++}$ such that

(2.4) \begin{gather} \|\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t} \mathrm{e}^{t{\widetilde{{{\boldsymbol{{B}}}}}}} - {\widetilde{{{\boldsymbol{{u}}}}}} {{\boldsymbol{{u}}}}^\top\| \leqslant C_1 \mathrm{e}^{-C_2 t} , \qquad \|\mathrm{e}^{t{\widetilde{{{\boldsymbol{{B}}}}}}}\| \leqslant C_3 \mathrm{e}^{s({\widetilde{{{\boldsymbol{{B}}}}}})t} , \qquad t \in \mathbb{R}_+ , \end{gather}

(2.5) \begin{gather} {\mathbb{E}}(\|{{\boldsymbol{{X}}}}_t\|) \leqslant C_4 \mathrm{e}^{s({\widetilde{{{\boldsymbol{{B}}}}}})t} , \qquad t \in \mathbb{R}_+ . \end{gather}

These Frobenius- and Perron-type results can be found, e.g., in Barczy and Pap [Reference Barczy and Pap10, Appendix A] and Barczy et al. [Reference Barczy, Palau and Pap8, (3.8)].

We will need the following dichotomy of the expectation of an irreducible multi-type CBI process.

Proof. For each $t \in \mathbb{R}_+$ , by (2.2), we have

\begin{equation*} {\mathbb{E}}({{\boldsymbol{{X}}}}_t) = \mathrm{e}^{t{\widetilde{{{\boldsymbol{{B}}}}}}} {\mathbb{E}}({{\boldsymbol{{X}}}}_0) + \int_0^t \mathrm{e}^{u{\widetilde{{{\boldsymbol{{B}}}}}}} \widetilde{{\boldsymbol{\beta}}} \, \mathrm{d} u , \qquad t \in \mathbb{R}_+^d . \end{equation*}

Since $\mathrm{e}^{u{\widetilde{{{\boldsymbol{{B}}}}}}} \in \mathbb{R}_{++}^{d\times d}$ for all $u \in \mathbb{R}_{++}$ , ${\mathbb{E}}({{\boldsymbol{{X}}}}_0) \in \mathbb{R}_+^d$ , and $\widetilde{{\boldsymbol{\beta}}} \in \mathbb{R}_+^d$ , we obtain the assertions.

We note that if $({{\boldsymbol{{X}}}}_t^{(1)})_{t\in\mathbb{R}_+}$ and $({{\boldsymbol{{X}}}}_t^{(2)})_{t\in\mathbb{R}_+}$ are multi-type CBI processes with parameters $(d, {{\boldsymbol{{c}}}}^{(1)}, {\boldsymbol{\beta}}, {{\boldsymbol{{B}}}}^{(1)}, \nu, \boldsymbol{\mu}^{(1)})$ and $(d, {{\boldsymbol{{c}}}}^{(2)}, {\boldsymbol{\beta}}, {{\boldsymbol{{B}}}}^{(2)}, \nu, \boldsymbol{\mu}^{(2)})$ , respectively, ${{\boldsymbol{{X}}}}_0^{(1)} \stackrel{{\mathrm{a.s.}}}{=} {{\boldsymbol{{X}}}}_0^{(2)}$ , and $({{\boldsymbol{{X}}}}_t^{(1)})_{t\in\mathbb{R}_+}$ is trivial, then $({{\boldsymbol{{X}}}}_t^{(2)})_{t\in\mathbb{R}_+}$ is also trivial.

Definition 2.5. Let $({{\boldsymbol{{X}}}}_t)_{t\in\mathbb{R}_+}$ be an irreducible multi-type CBI process with parameters $(d, {{\boldsymbol{{c}}}}, {\boldsymbol{\beta}}, {{\boldsymbol{{B}}}}, \nu, \boldsymbol{\mu})$ such that ${\mathbb{E}}(\|{{\boldsymbol{{X}}}}_0\|) < \infty$ and the moment condition (2.1) holds. Then $({{\boldsymbol{{X}}}}_t)_{t\in\mathbb{R}_+}$ is called

\begin{equation*} \begin{cases} subcritical & \text{if $s({\widetilde{{{\boldsymbol{{B}}}}}}) < 0$,} \\ critical & \text{if $s({\widetilde{{{\boldsymbol{{B}}}}}}) = 0$,} \\ supercritical & \text{if $s({\widetilde{{{\boldsymbol{{B}}}}}}) > 0$.} \end{cases} \end{equation*}

For the motivations for Definitions 2.3 and 2.5, see Barczy and Pap [Reference Barczy and Pap10, Section 3].

4. Proofs

Proof of Part (i) of Theorem 3.1. This statement is proved in Barczy et al. [Reference Barczy, Palau and Pap8, Theorem 3.1].

Proof of Part (iii) of Theorem 3.1. The proof is divided into three main steps. First, we decompose the process $(\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle)_{t\in\mathbb{R}_+}$ as the sum of a deterministic process and three square-integrable martingales. We show that the deterministic process goes to zero as $t\rightarrow\infty$ . For proving asymptotic normality of the martingales in question, we use Theorem 9, due to Crimaldi and Pratelli [Reference Crimaldi and Pratelli12, Theorem 2.2], which provides a set of sufficient conditions for the asymptotic normality of multivariate martingales. The proof is complete as soon as we show that the conditions (E.1) and (E.2) of Theorem 9 are satisfied. In the second and third steps, we prove that (E.1) and (E.2), respectively, are satisfied.

Step 1. For each $t \in \mathbb{R}_+$ , we have the representation

\begin{equation*} \mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle = Z_t^{(0,1)} + Z_t^{(2)} + Z_t^{(3,4)} + Z_t^{(5)} \end{equation*}

with

\begin{align*} Z_t^{(0,1)} &\,:\!=\, \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle + \langle{{\boldsymbol{{v}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle \int_0^t \mathrm{e}^{-\lambda u} \, \mathrm{d} u , \\ Z_t^{(2)} &\,:\!=\, \sum_{\ell=1}^d \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle \int_0^t \mathrm{e}^{-\lambda u} \sqrt{2 c_\ell X_{u,\ell}} \, \mathrm{d} W_{u,\ell} ,\\ Z_t^{(3,4)} &\,:\!=\, \sum_{\ell=1}^d \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, \widetilde{N}_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) , \\ Z_t^{(5)} &\,:\!=\, \int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle \, \widetilde{M}(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{r}}}}); \end{align*}

see Barczy et al. [Reference Barczy, Li and Pap7, Lemma 4.1] or Barczy et al. [Reference Barczy, Palau and Pap8, Lemma 2.7]. Thus, for each $t \in \mathbb{R}_+$ , we have

\begin{equation*} \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t/2} \langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle = \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)t/2} \bigl(Z_t^{(0,1)} + Z_t^{(2)} + Z_t^{(3,4)} + Z_t^{(5)}\bigr) . \end{equation*}

First, we show

(4.1) \begin{equation} \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)t/2} Z_t^{(0,1)} \stackrel{{\mathrm{a.s.}}}{\longrightarrow} 0 \qquad \text{as $t \to \infty$.} \end{equation}

Indeed, if $\lambda = 0$ , then

\begin{equation*} \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)t/2} Z_t^{(0,1)} = \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t/2} \left(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle + \langle{{\boldsymbol{{v}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle t\right) \stackrel{{\mathrm{a.s.}}}{\longrightarrow} 0 \qquad \text{as $t \to \infty$,} \end{equation*}

since $s({\widetilde{{{\boldsymbol{{B}}}}}}) \in \mathbb{R}_{++}$ . Otherwise, if ${\operatorname{Re}}(\lambda) \in \bigl(-\infty, \frac{1}{2} s({\widetilde{{{\boldsymbol{{B}}}}}})\bigr)$ and $\lambda \ne 0$ , then

\begin{align*} & \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)t/2} Z_t^{(0,1)} \\ &= \biggl(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle + \frac{\langle{{\boldsymbol{{v}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle}{\lambda}\biggr) \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t/2+\mathrm{i}{\operatorname{Im}}(\lambda)t} - \frac{\langle{{\boldsymbol{{v}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle}{\lambda} \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t/2} \stackrel{{\mathrm{a.s.}}}{\longrightarrow} 0 \end{align*}

as $t \to \infty$ .

For each $t \in \mathbb{R}_+$ , we have

\begin{equation*} \begin{pmatrix} {\operatorname{Re}}\bigl(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)t/2} \bigl(Z_t^{(2)} + Z_t^{(3,4)} + Z_t^{(5)}\bigr)\bigr) \\ {\operatorname{Im}}\bigl(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)t/2} \bigl(Z_t^{(2)} + Z_t^{(3,4)} + Z_t^{(5)}\bigr)\bigr) \end{pmatrix} = {{\boldsymbol{{Q}}}}(t) {{\boldsymbol{{M}}}}_t \end{equation*}

with

\begin{equation*} {{\boldsymbol{{Q}}}}(t) \,:\!=\, \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)t/2})\;\;\;\;\;\; & -{\operatorname{Im}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)t/2}) \\ {\operatorname{Im}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)t/2})\;\;\;\;\;\; & {\operatorname{Re}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)t/2}) \end{pmatrix} , \qquad t \in \mathbb{R}_+ , \end{equation*}

and

\begin{equation*} {{\boldsymbol{{M}}}}_t \,:\!=\, \begin{pmatrix} {\operatorname{Re}}\bigl(Z_t^{(2)} + Z_t^{(3,4)} + Z_t^{(5)}\bigr) \\ {\operatorname{Im}}\bigl(Z_t^{(2)} + Z_t^{(3,4)} + Z_t^{(5)}\bigr) \end{pmatrix} , \qquad t \in \mathbb{R}_+ . \end{equation*}

The assumption ${\operatorname{Re}}(\lambda) \in \bigl(-\infty, \frac{1}{2} s({\widetilde{{{\boldsymbol{{B}}}}}})\bigr)$ implies

\begin{equation*} {{\boldsymbol{{Q}}}}(t) = \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t/2} \begin{pmatrix} \cos({\operatorname{Im}}(\lambda)t)\;\;\;\;\;\; & -\sin({\operatorname{Im}}(\lambda)t) \\ \sin({\operatorname{Im}}(\lambda)t)\;\;\;\;\;\; & \cos({\operatorname{Im}}(\lambda)t) \end{pmatrix} \to {\boldsymbol{0}} \qquad \text{as $t \to \infty$.} \end{equation*}

For each $t \in \mathbb{R}_+$ , we can write ${{\boldsymbol{{M}}}}_t = {{\boldsymbol{{M}}}}_t^{(2)} + {{\boldsymbol{{M}}}}_t^{(3,4)} + {{\boldsymbol{{M}}}}_t^{(5)}$ with

\begin{equation*} {{\boldsymbol{{M}}}}_t^{(2)} \,:\!=\, \begin{pmatrix} {\operatorname{Re}}\bigl(Z_t^{(2)}\bigr) \\ {\operatorname{Im}}\bigl(Z_t^{(2)}\bigr) \end{pmatrix} , \qquad {{\boldsymbol{{M}}}}_t^{(3,4)} \,:\!=\, \begin{pmatrix} {\operatorname{Re}}\bigl(Z_t^{(3,4)}\bigr) \\ {\operatorname{Im}}\bigl(Z_t^{(3,4)}\bigr) \end{pmatrix} , \qquad {{\boldsymbol{{M}}}}_t^{(5)} \,:\!=\, \begin{pmatrix} {\operatorname{Re}}\bigl(Z_t^{(5)}\bigr) \\ {\operatorname{Im}}\bigl(Z_t^{(5)}\bigr) \end{pmatrix} . \end{equation*}

Note that under the moment condition (3.3), the stochastic processes $({{\boldsymbol{{M}}}}_t^{(2)})_{t\in\mathbb{R}_+}$ , $({{\boldsymbol{{M}}}}_t^{(3,4)})_{t\in\mathbb{R}_+}$ , and $({{\boldsymbol{{M}}}}_t^{(5)})_{t\in\mathbb{R}_+}$ are square-integrable martingales (see, e.g., Ikeda and Watanabe [Reference Ikeda and Watanabe16, pages 55 and 63]). One can also observe that, by the decomposition of $(\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle)_{t\in\mathbb{R}_+}$ given at the beginning of this step, $(\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle-\langle{{\boldsymbol{{v}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle \int_0^t \mathrm{e}^{-\lambda u} \, \mathrm{d} u)_{t\in\mathbb{R}_+}$ is a martingale with respect to the filtration $\sigma({{\boldsymbol{{X}}}}_u : u \in[0,t])$ , $t\in\mathbb{R}_+$ , which follows from Barczy et al. [Reference Barczy, Palau and Pap8, Lemma 2.6] as well.

The aim of the following discussion is to apply Theorem E.1 for the 2-dimensional martingale $({{\boldsymbol{{M}}}}_t)_{t\in\mathbb{R}_+}$ with the scaling ${{\boldsymbol{{Q}}}}(t)$ , $t \in \mathbb{R}_+$ .

Step 2. Now we prove that the condition (E.1) of Theorem E.1 holds for $({{\boldsymbol{{M}}}}_t)_{t\in\mathbb{R}_+}$ with the scaling ${{\boldsymbol{{Q}}}}(t)$ , $t \in \mathbb{R}_+$ . For each $t \in \mathbb{R}_+$ , by Theorem I.4.52 in Jacod and Shiryaev [Reference Jacod and Shiryaev17], we have

\begin{align*} [{{\boldsymbol{{M}}}}^{(2)}]_t &= \begin{pmatrix} [{\operatorname{Re}}\bigl(Z^{(2)}\bigr), {\operatorname{Re}}\bigl(Z^{(2)}\bigr)]_{t}\;\;\;\;\;\; & [{\operatorname{Re}}\bigl(Z^{(2)}\bigr), {\operatorname{Im}}\bigl(Z^{(2)}\bigr)]_t \\ [{\operatorname{Im}}\bigl(Z^{(2)}\bigr), {\operatorname{Re}}\bigl(Z^{(2)}\bigr)]_{t}\;\;\;\;\;\; & [{\operatorname{Im}}\bigl(Z^{(2)}\bigr), {\operatorname{Im}}\bigl(Z^{(2)}\bigr)]_t \end{pmatrix} \\ &= \begin{pmatrix} \langle{\operatorname{Re}}\bigl(Z^{(2)}\bigr), {\operatorname{Re}}\bigl(Z^{(2)}\bigr)\rangle_{t}\;\;\;\;\;\; & \langle{\operatorname{Re}}\bigl(Z^{(2)}\bigr), {\operatorname{Im}}\bigl(Z^{(2)}\bigr)\rangle_t \\ \langle{\operatorname{Im}}\bigl(Z^{(2)}\bigr), {\operatorname{Re}}\bigl(Z^{(2)}\bigr)\rangle_{t}\;\;\;\;\;\; & \langle{\operatorname{Im}}\bigl(Z^{(2)}\bigr), {\operatorname{Im}}\bigl(Z^{(2)}\bigr)\rangle_t \end{pmatrix} \\ &= 2 \sum_{\ell=1}^d c_\ell \int_0^t \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle) \\ {\operatorname{Im}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle) \end{pmatrix} \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle) \\ {\operatorname{Im}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle) \end{pmatrix}^\top X_{u,\ell} \, \mathrm{d} u , \end{align*}

since $({{\boldsymbol{{M}}}}_t^{(2)})_{t\in\mathbb{R}_+}$ is continuous, where $([{{\boldsymbol{{M}}}}^{(2)}]_t)_{t\in\mathbb{R}_+}$ and $(\langle{{\boldsymbol{{M}}}}^{(2)}\rangle_t)_{t\in\mathbb{R}_+}$ respectively denote the quadratic variation process and the predictable quadratic variation process of $({{\boldsymbol{{M}}}}^{(2)}_t)_{t\in\mathbb{R}_+}$ . Moreover, we have ${{\boldsymbol{{M}}}}_t^{(3,4)} = \sum_{\ell=1}^d {\widetilde{{{\boldsymbol{{Y}}}}}}_t^{(\ell)}$ with

\begin{equation*} {\widetilde{{{\boldsymbol{{Y}}}}}}_t^{(\ell)} \,:\!=\, \begin{pmatrix} {\operatorname{Re}}\bigl(\widetilde{Y}_t^{(\ell)}\bigr) \\ {\operatorname{Im}}\bigl(\widetilde{Y}_t^{(\ell)}\bigr) \end{pmatrix} , \qquad \widetilde{Y}_t^{(\ell)} \,:\!=\, \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, \widetilde{N}_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) \end{equation*}

for $t \in \mathbb{R}_+$ and $\ell \in \{1, \ldots, d\}$ . For each $t \in \mathbb{R}_+$ and $\ell \in \{1, \ldots, d\}$ , $({\widetilde{{{\boldsymbol{{Y}}}}}}_t^{(\ell)})_{t\in\mathbb{R}_+}$ is a square-integrable purely discontinuous martingale; see, e.g., Jacod and Shiryaev [Reference Jacod and Shiryaev17, Definition II.1.27 and Theorem II.1.33]. Hence, for each $t \in \mathbb{R}_+$ and $k, \ell \in \{1, \ldots, d\}$ , by Lemma I.4.51 in Jacod and Shiryaev [Reference Jacod and Shiryaev17], we have

\begin{equation*} [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)}, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}]_t = \sum_{s\in[0,t]} ({\widetilde{{{\boldsymbol{{Y}}}}}}_s^{(k)} - {\widetilde{{{\boldsymbol{{Y}}}}}}_{s-}^{(k)}) ({\widetilde{{{\boldsymbol{{Y}}}}}}_s^{(\ell)} - {\widetilde{{{\boldsymbol{{Y}}}}}}_{s-}^{(\ell)})^\top . \end{equation*}

Further, by the proof of Part (a) of Theorem II.1.33 in Jacod and Shiryaev [Reference Jacod and Shiryaev17], for each $t \in \mathbb{R}_+$ and $k \in \{1, \ldots, d\}$ ,

\begin{equation*} [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)}]_t = \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} \!\! \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \\ {\operatorname{Im}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \end{pmatrix}\! \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \\ {\operatorname{Im}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \end{pmatrix}^\top\!\! {\mathbb{1}}_{\{w\leqslant X_{u-,k}\}} \, N_k(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) . \end{equation*}

The aim of the following discussion is to show that for each $t \in \mathbb{R}_+$ and $k, \ell \in \{1, \ldots, d\}$ with $k \ne \ell$ , we have $[{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)}, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}]_t = {\boldsymbol{0}}$ almost surely. By the bilinearity of the quadratic variation process, for all $\varepsilon \in \mathbb{R}_{++}$ and $t \in \mathbb{R}_+$ , we have

(4.2) \begin{align} \begin{split} [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)}, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}]_t &= [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)}, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}]_t + [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)} , {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)} ]_t \\ &\phantom{=\;} + [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)} , {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)} ]_t + [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)} , {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)} ]_t , \end{split} \end{align}

where, for all $\varepsilon \in \mathbb{R}_{++}$ , $k \in \{1, \ldots, d\}$ and $t \in \mathbb{R}_+$ ,

\begin{align*} &{\widetilde{{{\boldsymbol{{Y}}}}}}_t^{(k,\varepsilon)} \,:\!=\, \begin{pmatrix} {\operatorname{Re}}\bigl(\widetilde{Y}_t^{(k,\varepsilon)}\bigr) \\[5pt] {\operatorname{Im}}\bigl(\widetilde{Y}_t^{(k, \varepsilon)}\bigr) \end{pmatrix} , \\ &\widetilde{Y}_t^{(k,\varepsilon)} \,:\!=\, \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|\geqslant \varepsilon\}} {\mathbb{1}}_{\{w\leqslant X_{u-,k}\}} \, \widetilde{N}_k(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) , \end{align*}

which is well-defined and square-integrable, since, by (2) and (3.8),

\begin{align*} &\int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{-2{\operatorname{Re}}(\lambda)u} |\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle|^2 {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|\geqslant \varepsilon\}} {\mathbb{E}}(X_{u,k}) \, \mathrm{d} u \, \mu_k(\mathrm{d}{{\boldsymbol{{z}}}}) \\ &\leqslant C_4 \|{{\boldsymbol{{v}}}}\|^2 \int_0^t \mathrm{e}^{(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))u} \, \mathrm{d} u \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^2 {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|\geqslant \varepsilon\}} \, \mu_k(\mathrm{d}{{\boldsymbol{{z}}}}) <\infty . \end{align*}

For each $\varepsilon \in \mathbb{R}_{++}$ , $t \in \mathbb{R}_+$ , and $k, \ell \in \{1, \ldots, d\}$ , we have

\begin{align*} [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)}, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}]_t & = \sum_{s\in[0,t]} ({\widetilde{{{\boldsymbol{{Y}}}}}}_s^{(k,\varepsilon)} - {\widetilde{{{\boldsymbol{{Y}}}}}}_{s-}^{(k,\varepsilon)}) ({\widetilde{{{\boldsymbol{{Y}}}}}}_s^{(\ell,\varepsilon)} - {\widetilde{{{\boldsymbol{{Y}}}}}}_{s-}^{(\ell,\varepsilon)})^\top \\ & = \sum_{s\in[0,t]} ({{\boldsymbol{{Y}}}}_s^{(k,\varepsilon)} - {{\boldsymbol{{Y}}}}_{s-}^{(k,\varepsilon)}) ({{\boldsymbol{{Y}}}}_s^{(\ell,\varepsilon)} - {{\boldsymbol{{Y}}}}_{s-}^{(\ell,\varepsilon)})^\top \end{align*}

with

\begin{align*} & {{\boldsymbol{{Y}}}}_t^{(k,\varepsilon)} \,:\!=\, \begin{pmatrix} {\operatorname{Re}}\bigl(Y_t^{(k,\varepsilon)}\bigr) \\ {\operatorname{Im}}\bigl(Y_t^{(k, \varepsilon)}\bigr) \end{pmatrix} , \\ & Y_t^{(k,\varepsilon)} \,:\!=\, \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|\geqslant \varepsilon\}} {\mathbb{1}}_{\{w\leqslant X_{u-,k}\}} \, N_k(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) , \end{align*}

where the first equality follows from the proof of Part (a) of Theorem II.1.33 in Jacod and Shiryaev [Reference Jacod and Shiryaev17], and the second equality follows from (2), Part (vi) of Definition 2.1, and (3.8), since

\begin{align*} &\int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{-{\operatorname{Re}}(\lambda) u} |\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle| {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|\geqslant \varepsilon\}} {\mathbb{E}}(X_{u,k}) \, \mathrm{d} u \,\mu_k(\mathrm{d}{{\boldsymbol{{z}}}}) \\ &\leqslant \frac{C_4\|{{\boldsymbol{{v}}}}\|}{\varepsilon} \int_0^t \mathrm{e}^{(s({\widetilde{{{\boldsymbol{{B}}}}}})-{\operatorname{Re}}(\lambda))u} \, \mathrm{d} u \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^2 \,\mu_k(\mathrm{d}{{\boldsymbol{{z}}}}) < \infty , \end{align*}

and hence we have

\begin{align*} \widetilde{Y}_t^{(k,\varepsilon)} = Y_t^{(k,\varepsilon)} - \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|\geqslant \varepsilon\}} {\mathbb{1}}_{\{w\leqslant X_{u-,k}\}} \, \mathrm{d} u \,\mu_k(\mathrm{d}{{\boldsymbol{{z}}}}) \, \mathrm{d} w . \end{align*}

For each $\varepsilon \in \mathbb{R}_{++}$ and $k \in \{1, \ldots, d\}$ , the jump times of $({{\boldsymbol{{Y}}}}_t^{(k,\varepsilon)})_{t\in\mathbb{R}_+}$ form a subset of the jump times of the Poisson process $(N_k([0,t] \times {\mathcal U}_d \times {\mathcal U}_1))_{t\in\mathbb{R}_+}$ . For each $k, \ell \in \{1, \ldots, d\}$ with $k \ne \ell$ , the Poisson processes $(N_k([0,t] \times {\mathcal U}_d \times {\mathcal U}_1))_{t\in\mathbb{R}_+}$ and $(N_\ell([0,t] \times {\mathcal U}_d \times {\mathcal U}_1))_{t\in\mathbb{R}_+}$ are independent, so they can jump simultaneously with probability zero; see, e.g., Revuz and Yor [Reference Revuz and Yor23, Chapter XII, Proposition 1.5]. Consequently, for each $\varepsilon \in \mathbb{R}_{++}$ , $t \in \mathbb{R}_+$ , and $k, \ell \in \{1, \ldots, d\}$ with $k \ne \ell$ , we have

\begin{equation*}[{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)}, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}]_t = {\boldsymbol{0}}\end{equation*}

almost surely.

Moreover, for each $t \in \mathbb{R}_+$ , $\varepsilon \in \mathbb{R}_{++}$ , $i, j \in \{1, 2\}$ and $k, \ell \in \{1, \ldots, d\}$ with $k \ne \ell$ , by the Kunita–Watanabe inequality, we have

\begin{align*} &\bigl|\langle {{\boldsymbol{{e}}}}_i, [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)} , {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}]_t {{\boldsymbol{{e}}}}_j \rangle\bigr| = \bigl|[\langle {{\boldsymbol{{e}}}}_i, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)} \rangle , \langle {{\boldsymbol{{e}}}}_j, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)} \rangle ]_t \bigr|\\ &\phantom{\bigl|\langle {{\boldsymbol{{e}}}}_i, [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)} } \leqslant [\langle {{\boldsymbol{{e}}}}_i , {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)} \rangle]_t^{1/2} \, [\langle {{\boldsymbol{{e}}}}_j , {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)}\rangle ]_t^{1/2} , \\ & \bigl|\langle {{\boldsymbol{{e}}}}_i, [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)} , {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}]_t {{\boldsymbol{{e}}}}_j \rangle \bigr| \leqslant [\langle {{\boldsymbol{{e}}}}_i, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)}\rangle ]_t^{1/2} \, [\langle {{\boldsymbol{{e}}}}_j, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)} \rangle ]_t^{1/2} , \\ & \bigl|\langle {{\boldsymbol{{e}}}}_i, [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)} , {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)} ]_t {{\boldsymbol{{e}}}}_j \rangle\bigr| \leqslant [\langle{{\boldsymbol{{e}}}}_i, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)} \rangle]_t^{1/2} \, [\langle{{\boldsymbol{{e}}}}_j, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)} \rangle]_t^{1/2} .\end{align*}

Hence it is enough to check that $[\langle {{\boldsymbol{{e}}}}_j, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)} \rangle]_t$ is stochastically bounded in $\varepsilon \in \mathbb{R}_{++}$ and

\begin{equation*} [\langle{{\boldsymbol{{e}}}}_j, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}\rangle]_t \stackrel{L_1}{\longrightarrow} 0 \qquad \text{as $\varepsilon \downarrow 0$} \end{equation*}

for all $t \in \mathbb{R}_+$ , $j \in \{1, 2\}$ , and $\ell \in \{1,\ldots, d\}$ . Indeed, in this case

\begin{align*} &\bigl|\langle {{\boldsymbol{{e}}}}_i, [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)} , {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}]_t {{\boldsymbol{{e}}}}_j \rangle\bigr| \stackrel{\mathbb{P}}{\longrightarrow} 0 \qquad \text{as $\varepsilon \downarrow 0$,}\\ &\bigl|\langle {{\boldsymbol{{e}}}}_i, [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)} , {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}]_t {{\boldsymbol{{e}}}}_j \rangle \bigr| \stackrel{\mathbb{P}}{\longrightarrow} 0 \qquad \text{as $\varepsilon \downarrow 0$,}\\ &\bigl|\langle {{\boldsymbol{{e}}}}_i, [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(k,\varepsilon)} , {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)} ]_t {{\boldsymbol{{e}}}}_j \rangle\bigr| \stackrel{\mathbb{P}}{\longrightarrow} 0 \qquad \text{as $\varepsilon \downarrow 0$,} \end{align*}

and, by (4.2), for each $t \in \mathbb{R}_+$ and $k, \ell \in \{1, \ldots, d\}$ with $k \ne \ell$ , we have

\begin{equation*}[{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)}, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}]_t = {\boldsymbol{0}}\end{equation*}

almost surely. By the proof of Part (a) of Theorem II.1.33 in Jacod and Shiryaev [Reference Jacod and Shiryaev17],

\begin{align*} [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}]_t = \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} & \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \\ {\operatorname{Im}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \end{pmatrix} \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \\ {\operatorname{Im}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \end{pmatrix}^\top\\ &\;\times{\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|\geqslant \varepsilon\}} {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, N_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) , \end{align*}

and

\begin{align*} [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}]_t = \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} & \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \\ {\operatorname{Im}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \end{pmatrix} \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \\ {\operatorname{Im}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \end{pmatrix}^\top \\ &\;\times {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|<\varepsilon\}} {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, N_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) . \end{align*}

Consequently, using that $\|{{\boldsymbol{{z}}}} {{\boldsymbol{{z}}}}^\top\| \leqslant \|{{\boldsymbol{{z}}}}\|^2$ , ${{\boldsymbol{{z}}}} \in \mathbb{R}^2$ , we have

\begin{equation*} \bigl|[\langle{{\boldsymbol{{e}}}}_j, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}\rangle]_t\bigr| \leqslant \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} |\mathrm{e}^{-\lambda u} \langle {{\boldsymbol{{v}}}},{{\boldsymbol{{z}}}}\rangle|^2 {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, N_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) \end{equation*}

for all $\varepsilon \in \mathbb{R}_{++}$ and $j\in\{1, 2\}$ , where the right-hand side is finite almost surely, since

\begin{align*} &{\mathbb{E}}\left( \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} |\mathrm{e}^{-\lambda u} \langle {{\boldsymbol{{v}}}},{{\boldsymbol{{z}}}}\rangle|^2 {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, N_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) \right) \\ & = \int_0^t \int_{{\mathcal U}_d} |\mathrm{e}^{-\lambda u} \langle {{\boldsymbol{{v}}}},{{\boldsymbol{{z}}}}\rangle|^2 {\mathbb{E}}(X_{u,\ell})\,\mathrm{d} u\,\mu_\ell(\mathrm{d} {{\boldsymbol{{z}}}}) \\ &\leqslant C_4 \|{{\boldsymbol{{v}}}}\|^2 \int_0^t \mathrm{e}^{(s({\widetilde{{{\boldsymbol{{B}}}}}}) -2{\operatorname{Re}}(\lambda))u}\,\mathrm{d} u \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^2 \,\mu_{\ell}(\mathrm{d}{{\boldsymbol{{z}}}}) <\infty. \end{align*}

Further,

\begin{align*} &{\mathbb{E}}\bigl(\bigl|[\langle{{\boldsymbol{{e}}}}_j, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}\rangle]_t\bigr|\bigr)\\ &\quad\leqslant {\mathbb{E}}\left(\int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} |\mathrm{e}^{-\lambda u} \langle {{\boldsymbol{{v}}}},{{\boldsymbol{{z}}}}\rangle|^2 {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|<\varepsilon\}} {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, N_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w)\right) \\ &\quad \leqslant \int_0^t \int_{{\mathcal U}_d} |\mathrm{e}^{-\lambda u} \langle {{\boldsymbol{{v}}}},{{\boldsymbol{{z}}}}\rangle|^2 {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|<\varepsilon\}} {\mathbb{E}}(X_{u,\ell})\,\mathrm{d} u\,\mu_\ell(\mathrm{d} {{\boldsymbol{{z}}}})\\ &\quad \leqslant C_4 \|{{\boldsymbol{{v}}}}\|^2 \int_0^t \mathrm{e}^{(s({\widetilde{{{\boldsymbol{{B}}}}}}) -2{\operatorname{Re}}(\lambda))u}\,\mathrm{d} u \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^2 {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|<\varepsilon\}}\,\mu_{\ell}(\mathrm{d}{{\boldsymbol{{z}}}}) \to 0 \end{align*}

as $\varepsilon \downarrow 0$ . Consequently, for each $t \in \mathbb{R}_+$ and $k, \ell \in \{1, \ldots, d\}$ with $k \ne \ell$ , we have

\begin{equation*}[{\widetilde{{{\boldsymbol{{Y}}}}}}^{(k)}, {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}]_t = {\boldsymbol{0}}\end{equation*}

almost surely.

In a similar way,

\begin{equation*} [{{\boldsymbol{{M}}}}^{(5)}]_t = \int_0^t \int_{{\mathcal U}_d} \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle) \\ {\operatorname{Im}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle) \end{pmatrix} \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle) \\ {\operatorname{Im}}(\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle) \end{pmatrix}^\top M(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{r}}}}) , \qquad t \in \mathbb{R}_+ , \end{equation*}

and $[{\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}, {{\boldsymbol{{M}}}}^{(5)}]_t = {\boldsymbol{0}}$ , $\ell \in \{1, \ldots, d\}$ , almost surely. Consequently, for each $t \in \mathbb{R}_+$ , we have $[{{\boldsymbol{{M}}}}^{(3,4)} + {{\boldsymbol{{M}}}}^{(5)}]_t = [{{\boldsymbol{{M}}}}^{(3,4)}]_t + [{{\boldsymbol{{M}}}}^{(5)}]_t$ with

\begin{equation*}[{{\boldsymbol{{M}}}}^{(3,4)}]_t = \sum_{\ell=1}^d [{\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}]_t.\end{equation*}

Since $({{\boldsymbol{{M}}}}^{(2)}_t)_{t\in\mathbb{R}_+}$ is a continuous martingale and $({{\boldsymbol{{M}}}}^{(3,4)}_t + {{\boldsymbol{{M}}}}^{(5)}_t)_{t\in\mathbb{R}_+}$ is a purely discontinuous martingale, by Corollary I.4.55 in Jacod and Shiryaev [Reference Jacod and Shiryaev17], we have $[{{\boldsymbol{{M}}}}^{(2)}, {{\boldsymbol{{M}}}}^{(3,4)} + {{\boldsymbol{{M}}}}^{(5)}]_t = {\boldsymbol{0}}$ , $t \in \mathbb{R}_+$ . Consequently,

\begin{equation*} [{{\boldsymbol{{M}}}}]_t = [{{\boldsymbol{{M}}}}^{(2)}]_t + [{{\boldsymbol{{M}}}}^{(3,4)}]_t + [{{\boldsymbol{{M}}}}^{(5)}]_t, \qquad t \in \mathbb{R}_+ . \end{equation*}

For each $t \in \mathbb{R}_+$ , we have

\begin{align*} {{\boldsymbol{{Q}}}}(t) [{{\boldsymbol{{M}}}}^{(2)}]_t {{\boldsymbol{{Q}}}}(t)^\top = 2 \sum_{\ell=1}^d c_\ell \int_0^t {{\boldsymbol{{f}}}}(t - \tau, {{\boldsymbol{{e}}}}_\ell) \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})\tau} X_{\tau,\ell} \, \mathrm{d} \tau \end{align*}

with

\begin{equation*} {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{z}}}}) \,:\!=\, \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)w/2} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \\[5pt] {\operatorname{Im}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)w/2} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \end{pmatrix} \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)w/2} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \\ {\operatorname{Im}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)w/2} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle) \end{pmatrix}^\top \end{equation*}

for $w \in \mathbb{R}_+$ and ${{\boldsymbol{{z}}}} \in \mathbb{R}^d$ . First, we show

(4.3) \begin{equation} {{\boldsymbol{{Q}}}}(t) [{{\boldsymbol{{M}}}}^{(2)}]_t {{\boldsymbol{{Q}}}}(t)^\top - 2 w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \sum_{\ell=1}^d c_\ell \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \int_0^t {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{e}}}}_\ell) \, \mathrm{d} w \stackrel{{\mathrm{a.s.}}}{\longrightarrow} {\boldsymbol{0}} \qquad \text{as $t \to \infty$.} \end{equation}

For each $t, T \in \mathbb{R}_+$ , we have

\begin{equation*} {{\boldsymbol{{Q}}}}(t + T) [{{\boldsymbol{{M}}}}^{(2)}]_{t+T} {{\boldsymbol{{Q}}}}(t + T)^\top - 2 w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \sum_{\ell=1}^d c_\ell \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \int_0^{t+T} {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{e}}}}_\ell) \, \mathrm{d} w = {\boldsymbol{\Delta}}_{t,T}^{(1)} + {\boldsymbol{\Delta}}_{t,T}^{(2)} \end{equation*}

with

\begin{align*} {\boldsymbol{\Delta}}_{t,T}^{(1)} &\,:\!=\, 2 \sum_{\ell=1}^d c_\ell \int_0^T {{\boldsymbol{{f}}}}(t + T - \tau, {{\boldsymbol{{e}}}}_\ell) (\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})\tau} X_{\tau,\ell} - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle) \, \mathrm{d} \tau , \\ {\boldsymbol{\Delta}}_{t,T}^{(2)} &\,:\!=\, 2 \sum_{\ell=1}^d c_\ell \int_T^{t+T} {{\boldsymbol{{f}}}}(t + T - \tau, {{\boldsymbol{{e}}}}_\ell) (\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})\tau} X_{\tau,\ell} - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle) \, \mathrm{d} \tau . \end{align*}

For each $t, T \in \mathbb{R}_+$ , we have

\begin{equation*} \|{\boldsymbol{\Delta}}_{t,T}^{(1)}\| \leqslant 2 \biggl(\sup_{\tau\in[0,T]} \|\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})\tau} {{\boldsymbol{{X}}}}_\tau - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} {\widetilde{{{\boldsymbol{{u}}}}}}\|\biggr) \sum_{\ell=1}^d c_\ell \int_0^T \|{{\boldsymbol{{f}}}}(t + T - \tau, {{\boldsymbol{{e}}}}_\ell)\| \, \mathrm{d} \tau , \end{equation*}

where

\begin{equation*}\sup_{\tau\in[0,T]} \|\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})\tau} {{\boldsymbol{{X}}}}_\tau - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} {\widetilde{{{\boldsymbol{{u}}}}}}\| < \infty\end{equation*}

almost surely, since $({{\boldsymbol{{X}}}}_t)_{t\in\mathbb{R}_+}$ has cÀdlÀg sample paths (by Theorem 4.6 in Barczy et al. [Reference Barczy, Li and Pap5]). Then, using that $\|{{\boldsymbol{{z}}}} {{\boldsymbol{{z}}}}^\top\| \leqslant \|{{\boldsymbol{{z}}}}\|^2$ , ${{\boldsymbol{{z}}}} \in \mathbb{R}^2$ , we have

(4.4) \begin{align} \begin{split} &\int_0^T \|{{\boldsymbol{{f}}}}(t + T - \tau, {{\boldsymbol{{e}}}}_\ell)\| \, \mathrm{d} \tau = \int_t^{t+T} \|{{\boldsymbol{{f}}}}(w, {{\boldsymbol{{e}}}}_\ell)\| \, \mathrm{d} w \leqslant \int_t^{t+T} |\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)w/2} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle|^2 \, \mathrm{d} w \\ &\leqslant \|{{\boldsymbol{{v}}}}\|^2 \int_t^\infty \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))w} \, \mathrm{d} w = \frac{\|{{\boldsymbol{{v}}}}\|^2}{s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda)} \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t} \to 0 \end{split} \end{align}

as $t \to \infty$ . Hence, for each $T \in \mathbb{R}_+$ , we obtain

\begin{equation*} \limsup_{t\to\infty} \|{\boldsymbol{\Delta}}_{t,T}^{(1)}\| = 0 \end{equation*}

almost surely. Moreover, for each $t, T \in \mathbb{R}_+$ , we have

\begin{equation*} \|{\boldsymbol{\Delta}}_{t,T}^{(2)}\| \leqslant 2 \biggl(\sup_{\tau\in[T,\infty)} \|\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})\tau} {{\boldsymbol{{X}}}}_\tau - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} {\widetilde{{{\boldsymbol{{u}}}}}}\|\biggr) \sum_{\ell=1}^d c_\ell \int_T^{t+T} \|{{\boldsymbol{{f}}}}(t + T - \tau, {{\boldsymbol{{e}}}}_\ell)\| \, \mathrm{d} \tau \end{equation*}

almost surely, where

(4.5) \begin{align} \begin{split} \int_T^{t+T} \|{{\boldsymbol{{f}}}}(t + T - \tau, {{\boldsymbol{{e}}}}_\ell)\| \, \mathrm{d} \tau &= \int_0^t \|{{\boldsymbol{{f}}}}(w, {{\boldsymbol{{e}}}}_\ell)\| \, \mathrm{d} w \leqslant\frac{\|{{\boldsymbol{{v}}}}\|^2}{s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda)} . \end{split} \end{align}

Consequently, for each $T \in \mathbb{R}_+$ , we obtain

\begin{align*} &\limsup_{t\to\infty} \biggl\|{{\boldsymbol{{Q}}}}(t) [{{\boldsymbol{{M}}}}^{(2)}]_t {{\boldsymbol{{Q}}}}(t)^\top - 2 w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \sum_{\ell=1}^d c_\ell \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \int_0^t {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{e}}}}_\ell) \, \mathrm{d} w\biggr\| \\ &= \limsup_{t\to\infty} \biggl\|{{\boldsymbol{{Q}}}}(t + T) [{{\boldsymbol{{M}}}}^{(2)}]_{t+T} {{\boldsymbol{{Q}}}}(t + T)^\top - 2 w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \sum_{\ell=1}^d c_\ell \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \int_0^{t+T} {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{e}}}}_\ell) \, \mathrm{d} w\biggr\| \\ &\leqslant \limsup_{t\to\infty} \|{\boldsymbol{\Delta}}_{t,T}^{(1)}\| + \limsup_{t\to\infty} \|{\boldsymbol{\Delta}}_{t,T}^{(2)}\| \\ &\leqslant \frac{2\|{{\boldsymbol{{v}}}}\|^2}{s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda)} \biggl(\sup_{\tau\in[T,\infty)} \|\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})\tau} {{\boldsymbol{{X}}}}_\tau - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} {\widetilde{{{\boldsymbol{{u}}}}}}\|\biggr) \sum_{\ell=1}^d c_\ell \end{align*}

almost surely. Letting $T \to \infty$ , by Theorem 3.3 in Barczy et al. [Reference Barczy, Palau and Pap8] (which can be used, since the moment condition (3.3) yields the moment condition (3.1) with $\lambda = s({\widetilde{{{\boldsymbol{{B}}}}}})$ ), we obtain (4.3). Moreover, $\int_0^t {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{e}}}}_\ell) \, \mathrm{d} w \to \int_0^\infty {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{e}}}}_\ell) \, \mathrm{d} w$ as $t \to \infty$ , since we have

\begin{equation*} \int_0^\infty \|{{\boldsymbol{{f}}}}(w, {{\boldsymbol{{e}}}}_\ell)\| \, \mathrm{d} w \leqslant \|{{\boldsymbol{{v}}}}\|^2 \int_0^\infty \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))w} \, \mathrm{d} w = \frac{\|{{\boldsymbol{{v}}}}\|^2}{s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda)} < \infty . \end{equation*}

Consequently,

(4.6) \begin{equation} {{\boldsymbol{{Q}}}}(t) [{{\boldsymbol{{M}}}}^{(2)}]_t {{\boldsymbol{{Q}}}}(t)^\top \stackrel{{\mathrm{a.s.}}}{\longrightarrow} 2 w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \sum_{\ell=1}^d c_\ell \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \int_0^\infty {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{e}}}}_\ell) \, \mathrm{d} w \qquad \text{as $t \to \infty$.} \end{equation}

Next, by Theorem 3.3 in Barczy et al. [Reference Barczy, Palau and Pap8], we show that

\begin{equation*} {{\boldsymbol{{Q}}}}(t) [{{\boldsymbol{{M}}}}^{(3,4)}]_t {{\boldsymbol{{Q}}}}(t)^\top - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \sum_{\ell=1}^d \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \int_0^t \int_{{\mathcal U}_d} {{\boldsymbol{{f}}}}(w,{{\boldsymbol{{z}}}}) \, \mathrm{d} w \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \stackrel{L_1}{\longrightarrow} {\boldsymbol{0}} \end{equation*}

as $t \to \infty$ . Since

(4.7) \begin{equation} {{\boldsymbol{{Q}}}}(t) [{{\boldsymbol{{M}}}}^{(3,4)}]_t {{\boldsymbol{{Q}}}}(t)^\top = \sum_{\ell=1}^d \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} {{\boldsymbol{{f}}}}(t - u,{{\boldsymbol{{z}}}}) \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})u} {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, N_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) , \end{equation}

it is enough to show that for each $\ell \in \{1, \ldots, d\}$ and $i, j \in \{1, 2\}$ , we have

(4.8) \begin{align} \begin{split} &\int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} f_{i,j}(t - u, {{\boldsymbol{{z}}}}) \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})u} {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, N_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) \\ &- w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \int_0^t \int_{{\mathcal U}_d} f_{i,j}(t - u, {{\boldsymbol{{z}}}}) \, \mathrm{d} u \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \stackrel{L_1}{\longrightarrow} 0 \qquad \text{as $t \to \infty$,} \end{split} \end{align}

where ${{\boldsymbol{{f}}}}(w,{{\boldsymbol{{z}}}})\,=\!:\,(f_{i,j}(w,{{\boldsymbol{{z}}}}))_{i,j\in\{1,2\}}$ , $w \in \mathbb{R}_+$ , ${{\boldsymbol{{z}}}} \in \mathbb{R}^d$ . For each $t \in \mathbb{R}_+$ , $\ell \in \{1, \ldots, d\}$ , and $i, j \in \{1, 2\}$ , we have

(4.9) \begin{equation} \begin{aligned} &{\mathbb{E}}\biggl(\biggl|\int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} f_{i,j}(t - u,{{\boldsymbol{{z}}}}) \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})u} {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, N_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) \\ &\phantom{{\mathbb{E}}\biggl(\biggl|} - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \int_0^t \int_{{\mathcal U}_d} f_{i,j}(t - u, {{\boldsymbol{{z}}}}) \, \mathrm{d} u \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}})\biggr|\biggr) \leqslant I_{t,1} + I_{t,2} , \end{aligned} \end{equation}

where

\begin{align*} I_{t,1} \,:\!=\, {\mathbb{E}}\biggl(\biggl|\int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} f_{i,j}(t - u,{{\boldsymbol{{z}}}}) \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})u} {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, \widetilde{N}_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w)\biggr|\biggr) \end{align*}

and

\begin{align*} I_{t,2} &\,:\!=\, {\mathbb{E}}\biggl(\biggl|\int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} f_{i,j}(t - u,{{\boldsymbol{{z}}}}) \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})u} {\mathbb{1}}_{\{w\leqslant X_{u,\ell}\}} \, \mathrm{d} u \,\mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}})\, \mathrm{d} w \\ &\phantom{\,:\!=\, {\mathbb{E}}\biggl(\biggl|} - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \int_0^t \int_{{\mathcal U}_d} f_{i,j}(t - u, {{\boldsymbol{{z}}}}) \, \mathrm{d} u \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}})\biggr|\biggr) \\ &= {\mathbb{E}}\biggl(\biggl|\int_0^t \int_{{\mathcal U}_d} f_{i,j}(t - u, {{\boldsymbol{{z}}}}) (\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})u} X_{u,\ell} - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle) \, \mathrm{d} w \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}})\biggr|\biggr) . \end{align*}

Here, for each $\ell \in \{1, \ldots, d\}$ and $i, j \in \{1 ,2\}$ , using Ikeda and Watanabe [Reference Ikeda and Watanabe16, page 63], (2), and the fact that $|{\operatorname{Re}}(a)| \leqslant |a|$ and $|{\operatorname{Im}}(a)| \leqslant |a|$ for each $a \in \mathbb{C}$ , we have

(4.10) \begin{equation} \begin{aligned} I_{t,1}^2 &\leqslant {\mathbb{E}}\Biggl(\Biggl| \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} f_{i,j}(t - u,{{\boldsymbol{{z}}}}) \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})u} {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, \widetilde{N}_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w)\Biggr|^2\Biggr) \\ &= \int_0^t \int_{{\mathcal U}_d} |f_{i,j}(t - u,{{\boldsymbol{{z}}}})|^2 \mathrm{e}^{-2s({\widetilde{{{\boldsymbol{{B}}}}}})u} {\mathbb{E}}(X_{u,\ell}) \, \mathrm{d} u \, \mu_\ell( \mathrm{d}{{\boldsymbol{{z}}}}) \\ &\leqslant \int_0^t \int_{{\mathcal U}_d} |\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}}) - 2\lambda)(t-u)/2} \langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle|^4 \mathrm{e}^{-2s({\widetilde{{{\boldsymbol{{B}}}}}})u} {\mathbb{E}}(\|{{\boldsymbol{{X}}}}_u\|) \, \mathrm{d} u \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \\ &\leqslant C_4 \|{{\boldsymbol{{v}}}}\|^4 \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^4 \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \, \mathrm{e}^{-2(s({\widetilde{{{\boldsymbol{{B}}}}}}) - 2{\operatorname{Re}}(\lambda))t} \int_0^t \mathrm{e}^{(s({\widetilde{{{\boldsymbol{{B}}}}}})-4{\operatorname{Re}}(\lambda))u} \, \mathrm{d} u \to 0 \end{aligned} \end{equation}

as $t \to \infty$ . Indeed, if $s({\widetilde{{{\boldsymbol{{B}}}}}}) \ne 4{\operatorname{Re}}(\lambda)$ , using that $2{\operatorname{Re}}(\lambda)<s({\widetilde{{{\boldsymbol{{B}}}}}})$ , we get

\begin{equation*} I_{t,1}^2 \leqslant C_4 \|{{\boldsymbol{{v}}}}\|^4 \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^4 \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \frac{\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t}-\mathrm{e}^{-2(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t}} {s({\widetilde{{{\boldsymbol{{B}}}}}})-4{\operatorname{Re}}(\lambda)} \to 0 \end{equation*}

as $t \to \infty$ , since $\int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^4 \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) < \infty$ . Otherwise, if $s({\widetilde{{{\boldsymbol{{B}}}}}}) = 4 {\operatorname{Re}}(\lambda)$ , then we obtain

\begin{equation*} I_{t,1}^2 \leqslant C_4 \|{{\boldsymbol{{v}}}}\|^4 \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^4 \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \, t \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t} \to 0 \qquad \text{as $t \to \infty$.} \end{equation*}

Further, for each $\ell\in\{1, \ldots, d\}$ , $i, j \in\{1, 2\}$ , and $t, T \in \mathbb{R}_+$ , we have

(4.11) \begin{equation} I_{t+T,2} \leqslant J_{t,T}^{(1)} + J_{t,T}^{(2)} \end{equation}

with

\begin{align*} J_{t,T}^{(1)} &\,:\!=\, {\mathbb{E}}\biggl(\biggl|\int_0^T \int_{{\mathcal U}_d} f_{i,j}(t + T - \tau, {{\boldsymbol{{z}}}}) (\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})\tau} X_{\tau,\ell} - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle) \, \mathrm{d} \tau \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}})\biggr|\biggr) , \\ J_{t,T}^{(2)} &\,:\!=\, {\mathbb{E}}\biggl(\biggl|\int_T^{t+T} \int_{{\mathcal U}_d} f_{i,j}(t + T - \tau, {{\boldsymbol{{z}}}}) (\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})\tau} X_{\tau,\ell} - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle) \, \mathrm{d} \tau \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}})\biggr|\biggr) . \end{align*}

By Theorem 3.3 in Barczy et al. [Reference Barczy, Palau and Pap8], we have

\begin{equation*}K \,:\!=\, \sup_{\tau\in\mathbb{R}_+} \!{\mathbb{E}}(\|\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})\tau} {{\boldsymbol{{X}}}}_\tau - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} {\widetilde{{{\boldsymbol{{u}}}}}}\|) < \infty,\end{equation*}

and hence, similarly as in (4.4), for any $T \in \mathbb{R}_+$ ,

\begin{align*} J_{t,T}^{(1)} &\leqslant K \int_0^T \int_{{\mathcal U}_d} |f_{i,j}(t + T - \tau, {{\boldsymbol{{z}}}})| \, \mathrm{d} \tau \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) = K \int_t^{t+T} \int_{{\mathcal U}_d} |f_{i,j}(w, {{\boldsymbol{{z}}}})| \, \mathrm{d} w \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \\ &\leqslant K \|{{\boldsymbol{{v}}}}\|^2 \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^2 \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \int_t^\infty \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))w} \, \mathrm{d} w \\ &= \frac{K\|{{\boldsymbol{{v}}}}\|^2}{s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda)} \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^2 \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \, \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}}) -2{\operatorname{Re}}(\lambda))t} \to 0 \end{align*}

as $t \to \infty$ . Further, similarly as in (4.5), for each $t, T \in \mathbb{R}_+$ ,

\begin{align*} J_{t,T}^{(2)} &\leqslant \sup_{\tau\in[T,\infty)} {\mathbb{E}}(|\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})\tau} X_{\tau,\ell} - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle|) \int_T^{t+T} \int_{{\mathcal U}_d} |f_{i,j}(t + T - \tau, {{\boldsymbol{{z}}}})| \, \mathrm{d} \tau \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \\ &\leqslant \sup_{\tau\in[T,\infty)} {\mathbb{E}}(|\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})\tau} X_{\tau,\ell} - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle|) \frac{\|{{\boldsymbol{{v}}}}\|^2}{s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda)} \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^2 \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) . \end{align*}

Consequently, for each $T \in \mathbb{R}_+$ , we obtain

\begin{align*} \limsup_{t\to\infty} I_{t,2} &= \limsup_{t\to\infty} I_{t+T,2} \leqslant \limsup_{t\to\infty} J_{t,T}^{(1)} + \limsup_{t\to\infty} J_{t,T}^{(2)} \\ &\leqslant \sup_{\tau\in[T,\infty)} {\mathbb{E}}(|\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})\tau} X_{\tau,\ell} - w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle|) \frac{\|{{\boldsymbol{{v}}}}\|^2}{s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda)} \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^2 \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) . \end{align*}

Letting $T \to \infty$ , by Theorem 3.3 in Barczy et al. [Reference Barczy, Palau and Pap8], we have $\lim_{t\to\infty} I_{t,2} = 0$ , as desired. All in all, $\lim_{t\to\infty} (I_{t,1} + I_{t,2}) = 0$ , yielding (4.8). Moreover, for each $\ell \in \{1, \ldots, d\}$ ,

\begin{equation*} \int_0^t \int_{{\mathcal U}_d} {{\boldsymbol{{f}}}}(t-u, {{\boldsymbol{{z}}}}) \, \mathrm{d} u \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) = \int_0^t \int_{{\mathcal U}_d} {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{z}}}}) \, \mathrm{d} w \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \to \int_0^\infty \int_{{\mathcal U}_d} {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{z}}}}) \, \mathrm{d} w \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \end{equation*}

as $t \to \infty$ , since we have

\begin{align*} \int_0^\infty \int_{{\mathcal U}_d} \|{{\boldsymbol{{f}}}}(w, {{\boldsymbol{{z}}}})\| \, \mathrm{d} w \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) &\leqslant \|{{\boldsymbol{{v}}}}\|^2 \int_0^\infty \int_{{\mathcal U}_d} \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))w} \|{{\boldsymbol{{z}}}}\|^2 \, \mathrm{d} w \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \\ &= \frac{\|{{\boldsymbol{{v}}}}\|^2}{s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda)} \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^2 \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) < \infty . \end{align*}

Consequently,

(4.12) \begin{equation} {{\boldsymbol{{Q}}}}(t) [{{\boldsymbol{{M}}}}^{(3,4)}]_t {{\boldsymbol{{Q}}}}(t)^\top \stackrel{L_1}{\longrightarrow} w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \sum_{\ell=1}^d \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \int_0^\infty \int_{{\mathcal U}_d} {{\boldsymbol{{f}}}}(w,{{\boldsymbol{{z}}}}) \, \mathrm{d} w \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \qquad \text{as $t \to \infty$.} \end{equation}

Further,

\begin{align*} {{\boldsymbol{{Q}}}}(t) [{{\boldsymbol{{M}}}}^{(5)}]_t {{\boldsymbol{{Q}}}}(t)^\top &= \int_0^t \int_{{\mathcal U}_d} {{\boldsymbol{{f}}}}(t - u,{{\boldsymbol{{r}}}}) \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})u} \, M(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{r}}}}) \stackrel{L_1}{\longrightarrow} {\boldsymbol{0}} \end{align*}

as $t \to \infty$ , since if ${\operatorname{Re}}(\lambda) \in (-\infty, \frac{1}{2}s({\widetilde{{{\boldsymbol{{B}}}}}}))$ , then

(4.13) \begin{equation} \begin{aligned} {\mathbb{E}}(\|{{\boldsymbol{{Q}}}}(t) [{{\boldsymbol{{M}}}}^{(5)}]_t {{\boldsymbol{{Q}}}}(t)^\top\|) &\leqslant {\mathbb{E}}\left(\int_0^t \int_{{\mathcal U}_d} \|{{\boldsymbol{{f}}}}(t - u, {{\boldsymbol{{r}}}}) \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})u}\| \, M(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{r}}}}) \right) \\ &= \int_0^t \int_{{\mathcal U}_d} \|{{\boldsymbol{{f}}}}(t - u,{{\boldsymbol{{r}}}}) \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})u} \| \, \mathrm{d} u\, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) \\ &\leqslant \int_0^t \int_{{\mathcal U}_d} \bigl|\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)(t-u)/2} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle\bigr|^2 \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})u} \, \mathrm{d} u \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) \\ &\leqslant \|{{\boldsymbol{{v}}}}\|^2 \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t} \int_0^t \mathrm{e}^{-2{\operatorname{Re}}(\lambda)u} \, \mathrm{d} u \int_{{\mathcal U}_d} \|{{\boldsymbol{{r}}}}\|^2 \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) \to 0 \end{aligned} \end{equation}

as $t \to \infty$ . Indeed, if ${\operatorname{Re}}(\lambda) \ne 0$ , then

\begin{equation*} \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t} \int_0^t \mathrm{e}^{-2{\operatorname{Re}}(\lambda)u} \, \mathrm{d} u = \frac{1}{2{\operatorname{Re}}(\lambda)} \Bigl(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t} - \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t}\Bigr) \to 0 \end{equation*}

as $t \to \infty$ , and if ${\operatorname{Re}}(\lambda) = 0$ , then

\begin{align*} \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t} \int_0^t \mathrm{e}^{-2{\operatorname{Re}}(\lambda)u} \, \mathrm{d} u = t \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t} \to 0 \end{align*}

as $t \to \infty$ . Consequently, by (4.6) and (4.12), we get

(4.14) \begin{align} \begin{split} {{\boldsymbol{{Q}}}}(t) [{{\boldsymbol{{M}}}}]_t {{\boldsymbol{{Q}}}}(t)^\top &\stackrel{\mathbb{P}}{\longrightarrow} 2 w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \sum_{\ell=1}^d c_\ell \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \int_0^\infty {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{e}}}}_\ell) \, \mathrm{d} w \\ &\phantom{\stackrel{\mathbb{P}}{\longrightarrow}\;} + w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \sum_{\ell=1}^d \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \int_0^\infty \int_{{\mathcal U}_d} {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{z}}}}) \, \mathrm{d} w \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \\ &= w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} {\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}} \qquad \text{as $t \to \infty$;} \end{split} \end{align}

hence the condition (E.1) of Theorem 9 holds. Indeed, for each $a \in \mathbb{C}$ , we have the identity

(4.15) \begin{equation} \begin{pmatrix} {\operatorname{Re}}(a) \\ {\operatorname{Im}}(a) \end{pmatrix} \begin{pmatrix} {\operatorname{Re}}(a) \\ {\operatorname{Im}}(a) \end{pmatrix}^\top = \frac{1}{2} |a|^2 {{\boldsymbol{{I}}}}_2 + \frac{1}{2} \begin{pmatrix} {\operatorname{Re}}(a^2)\;\;\;\;\;\; & {\operatorname{Im}}(a^2) \\ {\operatorname{Im}}(a^2)\;\;\;\;\;\; & - {\operatorname{Re}}(a^2) \end{pmatrix} . \end{equation}

Hence, for each $\ell \in \{1, \ldots, d\}$ , applying (4.15) with $a=\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})- 2\lambda)w/2} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle$ and from the linearity of the integral, we have

\begin{align*} &\int_0^\infty {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{e}}}}_\ell) \, \mathrm{d} w = \frac{1}{2} \int_0^\infty \begin{pmatrix} \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))w} |\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle|^2\;\;\;\;\; & 0 \\[5pt] 0\;\;\;\;\; & \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))w} |\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle|^2 \\ \end{pmatrix} \mathrm{d} w \\ &\phantom{\int_0^\infty {{\boldsymbol{{f}}}}(w, {{\boldsymbol{{e}}}}_\ell)\mathrm{d} w} + \frac{1}{2} \int_0^\infty \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)w} \langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle^2) \;\;\;\;\; & {\operatorname{Im}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)w} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle^2) \\ {\operatorname{Im}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)w} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle^2)\;\;\;\;\; & -{\operatorname{Re}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)w}\langle {{\boldsymbol{{v}}}},{{\boldsymbol{{e}}}}_\ell\rangle^2) \\ \end{pmatrix} \mathrm{d} w \\ &\qquad = \frac{|\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle|^2}{2(s^{^{^{^{}}}}\!({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))} {{\boldsymbol{{I}}}}_2 + \frac{1}{2} \begin{pmatrix} {\operatorname{Re}}\left(\frac{\langle{{\boldsymbol{{v}}}},{{\boldsymbol{{e}}}}_\ell\rangle^2}{s^{^{^{^{}}}}\!({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda}\right)\;\;\;\;\; & {\operatorname{Im}}\left(\frac{\langle{{\boldsymbol{{v}}}},{{\boldsymbol{{e}}}}_\ell\rangle^2} {s^{^{^{^{}}}}\!({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda}\right) \\[5pt] {\operatorname{Im}}\left(\frac{\langle{{\boldsymbol{{v}}}},{{\boldsymbol{{e}}}}_\ell\rangle^2}{s^{^{^{^{}}}}\!({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda}\right)\;\;\;\;\; & -{\operatorname{Re}}\left(\frac{\langle{{\boldsymbol{{v}}}},{{\boldsymbol{{e}}}}_\ell\rangle^2}{s^{^{^{^{}}}}\!({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda} \right) \end{pmatrix} , \end{align*}

and similarly

\begin{align*} &\int_0^\infty \int_{{\mathcal U}_d} {{\boldsymbol{{f}}}}(w,{{\boldsymbol{{z}}}}) \, \mathrm{d} w \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \\ &= \frac{\int_{{\mathcal U}_d}|\langle{{\boldsymbol{{v}}}},{{\boldsymbol{{z}}}} \rangle|^2 \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}})} {2(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))}{{\boldsymbol{{I}}}}_2 + \frac{1}{2} \begin{pmatrix} {\operatorname{Re}}\left(\frac{\int_{{\mathcal U}_d}\langle{{\boldsymbol{{v}}}},{{\boldsymbol{{z}}}}\rangle^2\,\mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}})} {s^{^{^{^{}}}}\!({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda}\right)\;\;\;\;\; & {\operatorname{Im}}\left(\frac{\int_{{\mathcal U}_d}\langle{{\boldsymbol{{v}}}},{{\boldsymbol{{z}}}}\rangle^2\,\mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}})} {s^{^{^{^{}}}}\!({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda}\right) \\[5pt] {\operatorname{Im}}\left(\frac{\int_{{\mathcal U}_d}\langle{{\boldsymbol{{v}}}},{{\boldsymbol{{z}}}}\rangle^2\,\mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}})} {s^{^{^{^{}}}}\!({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda}\right) & -{\operatorname{Re}}\left(\frac{\int_{{\mathcal U}_d}\langle{{\boldsymbol{{v}}}},{{\boldsymbol{{z}}}}\rangle^2\,\mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}})} {s^{^{^{^{}}}}\!({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda}\right) \end{pmatrix} , \end{align*}

yielding (4.14). Note that $ {\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}}$ is non-negative definite irrespective of $\widetilde{{\boldsymbol{\beta}}} \ne {\boldsymbol{0}}$ or $\widetilde{{\boldsymbol{\beta}}} = {\boldsymbol{0}}$ , since ${{\boldsymbol{{c}}}} \in \mathbb{R}_+^d$ , ${\widetilde{{{\boldsymbol{{u}}}}}} \in \mathbb{R}_{++}^d$ , and ${{\boldsymbol{{f}}}}(w,{{\boldsymbol{{z}}}})$ is non-negative definite for any $w \in \mathbb{R}_+$ and ${{\boldsymbol{{z}}}} \in \mathbb{R}^d$ .

Step 3. Now we turn to proving that the condition (E.2) of Theorem 9 holds for $({{\boldsymbol{{M}}}}_t)_{t\in\mathbb{R}_+}$ with the scaling ${{\boldsymbol{{Q}}}}(t)$ , $t \in \mathbb{R}_+$ ; that is,

\begin{equation*} {\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|{{\boldsymbol{{Q}}}}(t) ({{\boldsymbol{{M}}}}_u - {{\boldsymbol{{M}}}}_{u-})\|\biggr) \to 0 \qquad \text{as $t \to \infty$.} \end{equation*}

Since $({{\boldsymbol{{M}}}}^{(2)}_t)_{t\in\mathbb{R}_+}$ has continuous sample paths, for each $t \in \mathbb{R}_+$ we have

(4.16) \begin{equation} \begin{aligned} &\sup_{u\in[0,t]} \|{{\boldsymbol{{Q}}}}(t) ({{\boldsymbol{{M}}}}_u - {{\boldsymbol{{M}}}}_{u-})\| = \sup_{u\in[0,t]} \|{{\boldsymbol{{Q}}}}(t) ({{\boldsymbol{{M}}}}^{(3,4)}_u - {{\boldsymbol{{M}}}}^{(3,4)}_{u-}) + {{\boldsymbol{{Q}}}}(t) ({{\boldsymbol{{M}}}}^{(5)}_u - {{\boldsymbol{{M}}}}^{(5)}_{u-})\|\\[5pt] &\leqslant \|{{\boldsymbol{{Q}}}}(t)\| \sum_{\ell=1}^d \sup_{u\in[0,t]} \|{\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_u - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_{u-}\| + \|{{\boldsymbol{{Q}}}}(t)\| \sup_{u\in[0,t]} \|{{\boldsymbol{{M}}}}^{(5)}_u - {{\boldsymbol{{M}}}}^{(5)}_{u-}\| \end{aligned} \end{equation}

almost surely. Since ${{\boldsymbol{{Q}}}}(t) {{\boldsymbol{{Q}}}}(t)^\top = \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t} {{\boldsymbol{{I}}}}_2$ , $t \in \mathbb{R}_+$ , we have $\|{{\boldsymbol{{Q}}}}(t)\| = \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t/2}$ , $t \in \mathbb{R}_+$ . Hence it is enough to show that

(4.17) \begin{align} \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t/2} {\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|{\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_u - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_{u-}\|\biggr) \to 0 \qquad \text{as $t \to \infty$} \end{align}

for every $\ell \in \{1, \ldots, d\}$ , and

(4.18) \begin{align} \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t/2} {\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|{{\boldsymbol{{M}}}}^{(5)}_u - {{\boldsymbol{{M}}}}^{(5)}_{u-}\|\biggr) \to 0 \qquad \text{as $t \to \infty$.} \end{align}

First, we prove (4.17) for each $\ell \in \{1, \ldots, d\}$ . By the Cauchy–Schwarz inequality, for each $\varepsilon \in \mathbb{R}_{++}$ , $t \in \mathbb{R}_+$ , and $\ell \in \{1, \ldots, d\}$ , we have

(4.19) \begin{align} \begin{split} &\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t/2} {\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|{\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_u - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_{u-}\|\biggr) \\ &\leqslant \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t/2} {\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|{\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_u - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_{u-}\| \biggr) \\ &\phantom{\leqslant\;} + \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t/2} {\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|({\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_u - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_{u}) - ({\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_{u-} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_{u-})\|\biggr)\\ &\leqslant \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t/2} \Biggl({\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|{\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_u - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_{u-}\|^4\biggr) \Biggr)^{1/4} \\ &\phantom{\leqslant\;} + \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t/2} \Biggl({\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|({\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_u - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_{u} ) - ({\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_{u-} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_{u-})\|^2\biggr) \Biggr)^{1/2} . \end{split} \end{align}

Here, by (2), for each $\varepsilon \in \mathbb{R}_{++}$ , $t \in \mathbb{R}_+$ , and $\ell \in \{1, \ldots, d\}$ , we have

(4.20) \begin{align} \begin{split} &{\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|{\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_u - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_{u-}\|^4\biggr) = {\mathbb{E}}\biggl(\sup_{u\in[0,t]} |\widetilde{Y}^{(\ell,\varepsilon)}_u - \widetilde{Y}^{(\ell,\varepsilon)}_{u-}|^4\biggr)\\ & = {\mathbb{E}}\biggl(\sup_{u\in[0,t]} |Y^{(\ell,\varepsilon)}_u - Y^{(\ell,\varepsilon)}_{u-}|^4\biggr)\\ & \leqslant {\mathbb{E}}\Biggl(\sum_{u\in[0,t]} |Y^{(\ell,\varepsilon)}_u - Y^{(\ell,\varepsilon)}_{u-}|^4\Biggr) \\ &= {\mathbb{E}}\biggl(\int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} |\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|\geqslant \varepsilon\}} {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}}|^4 \, N_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w)\biggr)\\ & = \int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{-4{\operatorname{Re}}(\lambda)u} |\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle|^4 {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|\geqslant \varepsilon\}} {\mathbb{E}}(X_{u,\ell}) \, \mathrm{d} u \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \\ & \leqslant C_4 \|{{\boldsymbol{{v}}}}\|^4 \int_0^t \mathrm{e}^{(s({\widetilde{{{\boldsymbol{{B}}}}}})-4{\operatorname{Re}}(\lambda))u} \, \mathrm{d} u \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^4 {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|\geqslant \varepsilon\}} \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) . \end{split} \end{align}

Hence, by (4.10) and $2{\operatorname{Re}}(\lambda)<s({\widetilde{{{\boldsymbol{{B}}}}}})$ , for each $\varepsilon \in \mathbb{R}_{++}$ and $\ell \in \{1, \ldots, d\}$ , we get

(4.21) \begin{align} \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t/2} \Biggl({\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|{\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_u - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_{u-}\|^4\biggr)\Biggr)^{1/4} \to 0 \qquad \text{as $t \to \infty$.} \end{align}

Further, since

\begin{equation*} {\widetilde{{{\boldsymbol{{Y}}}}}}_t^{(\ell)} - {\widetilde{{{\boldsymbol{{Y}}}}}}_t^{(\ell,\varepsilon)} = \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|<\varepsilon\}} {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, \widetilde{N}_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w), \qquad t \in \mathbb{R}_+ , \end{equation*}

by the proof of Part (a) of Theorem II.1.33 in Jacod and Shiryaev [Reference Jacod and Shiryaev17], we get

(4.22) \begin{align} \begin{split} &{\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|({\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_u - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_{u}) - ({\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_{u-} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_{u-})\|^2\biggr) \\[5pt] &\leqslant {\mathbb{E}}\biggl(\sum_{u\in[0,t]} \|({\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_u - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_{u}) - ({\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_{u-} - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell,\varepsilon)}_{u-})\|^2\biggr) \\[5pt] &= {\mathbb{E}}\Biggl(\int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} |\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle|^2 {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|<\varepsilon\}} {\mathbb{1}}_{\{w\leqslant X_{u,\ell}\}} \, N_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w)\Biggr) \\[5pt] &\leqslant C_4 \|{{\boldsymbol{{v}}}}\|^2 \int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda)) u} \|{{\boldsymbol{{z}}}}\|^2 {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|<\varepsilon\}} \, \mu_{\ell}(\mathrm{d} {{\boldsymbol{{z}}}}) \\[5pt] &\leqslant C_4 \|{{\boldsymbol{{v}}}}\|^2 \frac{\mathrm{e}^{(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t}} {s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda)} \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^2 {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|<\varepsilon\}} \, \mu_{\ell}(\mathrm{d} {{\boldsymbol{{z}}}}) . \end{split} \end{align}

Hence, by (4.19) and (4.21), for all $\varepsilon \in \mathbb{R}_{++}$ and $\ell \in \{1, \ldots, d\}$ , we have

\begin{align*} \limsup_{t\to\infty} \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t/2} &{\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|{\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_u - {\widetilde{{{\boldsymbol{{Y}}}}}}^{(\ell)}_{u-}\|\biggr) \\[5pt] &\leqslant \Biggl(\frac{C_4\|{{\boldsymbol{{v}}}}\|^2} {s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda)} \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^2 {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|<\varepsilon\}} \, \mu_{\ell}(\mathrm{d} {{\boldsymbol{{z}}}}) \Biggr)^{1/2} , \end{align*}

which tends to 0 as $\varepsilon \downarrow 0$ by (3.8). Hence we conclude (4.17) for each $\ell \in \{1, \ldots, d\}$ .

Next, we prove (4.18). By the Cauchy–Schwarz inequality, for each $t \in \mathbb{R}_+$ , we have

(4.23) \begin{align} \begin{split} &{\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|{{\boldsymbol{{M}}}}^{(5)}_u - {{\boldsymbol{{M}}}}^{(5)}_{u-}\|\biggr)\\[5pt] &\leqslant \Biggl({\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|{{\boldsymbol{{M}}}}^{(5)}_u - {{\boldsymbol{{M}}}}^{(5)}_{u-}\|^2\biggr)\!\Biggr)^{1/2}\! = \Biggl({\mathbb{E}}\biggl(\sup_{u\in[0,t]} |Z^{(5)}_u - Z^{(5)}_{u-}|^2\biggr)\Biggr)^{1/2} , \end{split} \end{align}

so it is enough to prove that

\begin{equation*} \mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda))t} {\mathbb{E}}\biggl(\sup_{u\in[0,t]} |Z^{(5)}_u - Z^{(5)}_{u-}|^2\biggr) \to 0 \qquad \text{as $t \to \infty$.} \end{equation*}

Since $\int_{{\mathcal U}_d} \|{{\boldsymbol{{r}}}}\| \, \nu(\mathrm{d} {{\boldsymbol{{r}}}}) < \infty$ , for each $t \in \mathbb{R}_+$ we have

\begin{equation*} Z^{(5)}_t = Z^*_t - \int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle \, \mathrm{d} u \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) \end{equation*}

with $Z^*_t \,:\!=\, \int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle \, M(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{r}}}})$ ; therefore

(4.24) \begin{equation} \begin{aligned} &{\mathbb{E}}\biggl(\sup_{u\in[0,t]} |Z^{(5)}_u - Z^{(5)}_{u-}|^2\biggr) = {\mathbb{E}}\biggl(\sup_{u\in[0,t]} |Z^*_u - Z^*_{u-}|^2\biggr) \leqslant {\mathbb{E}}\Biggl(\sum_{u\in[0,t]} |Z^*_u - Z^*_{u-}|^2\Biggr) \\[5pt] &= {\mathbb{E}}\biggl(\int_0^t \int_{{\mathcal U}_d} |\mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle|^2 \, M(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{r}}}})\biggr) = \int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{-2{\operatorname{Re}}(\lambda)u} |\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle|^2 \, \mathrm{d} u \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) \\[5pt] &\leqslant \|{{\boldsymbol{{v}}}}\|^2 \int_0^t \mathrm{e}^{-2{\operatorname{Re}}(\lambda)u} \, \mathrm{d} u \int_{{\mathcal U}_d} \|{{\boldsymbol{{r}}}}\|^2 \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}), \end{aligned} \end{equation}

and so by (4.13), we conclude (4.18). Consequently, by Theorem E.1, we obtain

\begin{equation*} {{\boldsymbol{{Q}}}}(t) {{\boldsymbol{{M}}}}_t \stackrel{{\mathcal D}}{\longrightarrow} (w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} {\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}})^{1/2} {{\boldsymbol{{N}}}} \qquad \text{as $t \to \infty$,} \end{equation*}

where ${{\boldsymbol{{N}}}}$ is a 2-dimensional random vector with ${{\boldsymbol{{N}}}} \stackrel{{\mathcal D}}{=} {\mathcal N}_2({\boldsymbol{0}}, {{\boldsymbol{{I}}}}_2)$ independent of $w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} {\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}}$ . Clearly, $(w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} {\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}})^{1/2} {{\boldsymbol{{N}}}} = \sqrt{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}} \, {\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}}^{1/2} {{\boldsymbol{{N}}}} \stackrel{{\mathcal D}}{=} \sqrt{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}} {{\boldsymbol{{Z}}}}_{{\boldsymbol{{v}}}}$ . By the decomposition

\begin{equation*} \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t/2} \begin{pmatrix} {\operatorname{Re}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \\ {\operatorname{Im}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \end{pmatrix} = \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)t/2} Z_t^{(0,1)}) \\ {\operatorname{Im}}(\mathrm{e}^{-(s({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda)t/2} Z_t^{(0,1)}) \end{pmatrix} + {{\boldsymbol{{Q}}}}(t) {{\boldsymbol{{M}}}}_t , \qquad t \in \mathbb{R}_+ , \end{equation*}

the convergence (4.1), and Slutsky’s lemma (see, e.g., van der Vaart [Reference Van der Vaart24, Lemma 2.8]), we obtain (3.6).

Proof of Part (ii) of Theorem 3.1. This is similar to the proof of Part (iii) of Theorem 3. For details, see our ArXiv preprint [Reference Barczy, Palau and Pap9].

Proof of Theorem 3.2. First, suppose that the conditions (i) and (ii) hold. In the special case of ${{\boldsymbol{{X}}}}_0 \stackrel{{\mathrm{a.s.}}}{=} {\boldsymbol{0}}$ , applying Lemma A.1 with $T = 1$ , we have ${{\boldsymbol{{X}}}}_{t+1} \stackrel{{\mathcal D}}{=} {{\boldsymbol{{X}}}}_t^{(1)} + {{\boldsymbol{{X}}}}_t^{(2,1)}$ for each $t \in \mathbb{R}_+$ , where $({{\boldsymbol{{X}}}}_s^{(1)})_{s\in\mathbb{R}_+}$ and $({{\boldsymbol{{X}}}}_s^{(2,1)})_{s\in\mathbb{R}_+}$ are independent multi-type CBI processes with ${{\boldsymbol{{X}}}}_0^{(1)} \stackrel{{\mathrm{a.s.}}}{=} {\boldsymbol{0}}$ , ${{\boldsymbol{{X}}}}_0^{(2,1)} \stackrel{{\mathcal D}}{=} {{\boldsymbol{{X}}}}_1$ , and with parameters $(d, {{\boldsymbol{{c}}}}, {\boldsymbol{\beta}}, {{\boldsymbol{{B}}}}, \nu, \boldsymbol{\mu})$ and $(d, {{\boldsymbol{{c}}}}, {\boldsymbol{0}}, {{\boldsymbol{{B}}}}, 0, \boldsymbol{\mu})$ , respectively. Without loss of generality, we may and do suppose that $({{\boldsymbol{{X}}}}_s)_{s\in\mathbb{R}_+}$ , $({{\boldsymbol{{X}}}}_s^{(1)})_{s\in\mathbb{R}_+}$ , and $({{\boldsymbol{{X}}}}_s^{(2,1)})_{s\in\mathbb{R}_+}$ are independent. Then, for each $t \in \mathbb{R}_+$ , we have

\begin{equation*}\mathrm{e}^{-\lambda(t+1)} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_{t+1}\rangle \stackrel{{\mathcal D}}{=} \mathrm{e}^{-\lambda} (\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t^{(1)}\rangle) + \mathrm{e}^{-\lambda} (\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t^{(2,1)}\rangle).\end{equation*}

By (3.2), we obtain

\begin{equation*}w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}} \stackrel{{\mathcal D}}{=} \mathrm{e}^{-\lambda} w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}^{(1)} + \mathrm{e}^{-\lambda} w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,1)}}^{(2,1)},\end{equation*}

where

\begin{equation*}w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}^{(1)}, \qquad w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,1)}}^{(2,1)}\end{equation*}

respectively denote the almost sure limits of $\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t^{(1)}\rangle$ and $\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t^{(2,1)}\rangle$ as $t \to \infty$ . Since, for each $t \in \mathbb{R}_+$ , we have ${{\boldsymbol{{X}}}}_t^{(1)} \stackrel{{\mathcal D}}{=} {{\boldsymbol{{X}}}}_t$ , we conclude $w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}^{(1)} \stackrel{{\mathcal D}}{=} w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}$ . The independence of $({{\boldsymbol{{X}}}}_s)_{s\in\mathbb{R}_+}$ and $({{\boldsymbol{{X}}}}_s^{(2,1)})_{s\in\mathbb{R}_+}$ implies the independence of $w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}$ and $w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,1)}}^{(2,1)}$ ; hence

\begin{equation*}w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}} \stackrel{{\mathcal D}}{=} \mathrm{e}^{-\lambda} w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}} + \mathrm{e}^{-\lambda} w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,1)}}^{(2,1)}.\end{equation*}

Taking the real and imaginary parts, we get

\begin{align*} \begin{pmatrix} {\operatorname{Re}}(w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}) \\ {\operatorname{Im}}(w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}) \end{pmatrix} &\stackrel{{\mathcal D}}{=} \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-\lambda})\;\;\;\;\; & - {\operatorname{Im}}(\mathrm{e}^{-\lambda}) \\ {\operatorname{Im}}(\mathrm{e}^{-\lambda})\;\;\;\;\; & {\operatorname{Re}}(\mathrm{e}^{-\lambda}) \end{pmatrix}\! \begin{pmatrix} {\operatorname{Re}}(w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}) \\ {\operatorname{Im}}(w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}) \end{pmatrix}\\ &\phantom{\stackrel{{\mathcal D}}{=}\;} + \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-\lambda})\;\;\;\;\; & - {\operatorname{Im}}(\mathrm{e}^{-\lambda}) \\ {\operatorname{Im}}(\mathrm{e}^{-\lambda})\;\;\;\;\; & {\operatorname{Re}}(\mathrm{e}^{-\lambda}) \end{pmatrix}\! \begin{pmatrix} {\operatorname{Re}}(w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,1)}}^{(2,1)}) \\[5pt] {\operatorname{Im}}(w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,1)}}^{(2,1)}) \end{pmatrix} \\ &=\!:\;{\boldsymbol{{A}}} \begin{pmatrix} {\operatorname{Re}}(w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}) \\ {\operatorname{Im}}(w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}) \end{pmatrix} + {\boldsymbol{{A}}} {{\boldsymbol{{C}}}} , \end{align*}

which is a 2-dimensional stochastic fixed point equation. We are going to apply Corollary C.1. We have $\det({\boldsymbol{{A}}}) = ({\operatorname{Re}}(\mathrm{e}^{-\lambda}))^2 + ({\operatorname{Im}}(\mathrm{e}^{-\lambda}))^2 = |\mathrm{e}^{-\lambda}|^2 = \mathrm{e}^{-2{\operatorname{Re}}(\lambda)} \ne 0$ . The eigenvalues of the matrix ${\boldsymbol{{A}}}$ are $\mathrm{e}^{-\lambda}$ and $\mathrm{e}^{-\overline{\lambda}}$ , so the spectral radius of ${\boldsymbol{{A}}}$ is $r({\boldsymbol{{A}}}) = \mathrm{e}^{-{\operatorname{Re}}(\lambda)} \in (0, 1)$ . Next we check that ${\boldsymbol{{A}}} {{\boldsymbol{{C}}}}$ is not deterministic. Suppose that, on the contrary, ${\boldsymbol{{A}}} {{\boldsymbol{{C}}}}$ is deterministic. Then

\begin{equation*}w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,1)}}^{(2,1)}\end{equation*}

is deterministic, since ${\boldsymbol{{A}}}$ is invertible. By Lemma 2.6 in Barczy et al. [Reference Barczy, Palau and Pap8], the process $(\mathrm{e}^{-s{\widetilde{{{\boldsymbol{{B}}}}}}} {{\boldsymbol{{X}}}}_s^{(2,1)})_{s\in\mathbb{R}_+}$ is a d-dimensional martingale with respect to the filtration ${\mathcal F}^{{{\boldsymbol{{X}}}}^{(2,1)}}_s \,:\!=\, \sigma({{\boldsymbol{{X}}}}_u^{(2,1)} : u \in [0, s])$ , $s \in \mathbb{R}_+$ ; hence $(\mathrm{e}^{-\lambda s} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_s^{(2,1)}\rangle)_{s\in\mathbb{R}_+}$ is a complex martingale with respect to the same filtration. By (3.2), we have

\begin{equation*}\mathrm{e}^{-\lambda s} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_s^{(2,1)} \rangle \to w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,1)}}^{(2,1)}\end{equation*}

as $s \to \infty$ in $L_1$ and almost surely, so

\begin{equation*}\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0^{(2,1)}\rangle = {\mathbb{E}}(w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,1)}}^{(2,1)} \,|\, {\mathcal F}_0^{{{\boldsymbol{{X}}}}^{(2,1)}}) = w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,1)}}^{(2,1)}\end{equation*}

almost surely; see, e.g., Karatzas and Shreve [Reference Karatzas and Shreve20, Chapter I, Problem 3.20]. Thus $\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0^{(2,1)}\rangle$ is deterministic as well. Then $\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_1\rangle$ is also deterministic since ${{\boldsymbol{{X}}}}_1 \stackrel{{\mathcal D}}{=} {{\boldsymbol{{X}}}}_0^{(2,1)}$ . However, applying Lemma 6 for the process $({{\boldsymbol{{X}}}}_s)_{s\in\mathbb{R}_+}$ , we obtain that $\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_1\rangle$ is not deterministic, since the condition (i) of this theorem implies that the process $({{\boldsymbol{{X}}}}_s)_{s\in\mathbb{R}_+}$ is non-trivial, and the condition (ii) of this theorem yields that the condition (ii)/(b) of Lemma 6 does not hold. Thus we get a contradiction, and we conclude that ${\boldsymbol{{A}}} {{\boldsymbol{{C}}}}$ is not deterministic. Moreover, we have

\begin{equation*}{\mathbb{E}}(\|{{\boldsymbol{{C}}}}\|) = {\mathbb{E}}(|w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,1)}}^{(2,1)}|) < \infty;\end{equation*}

see (3.2). Applying Corollary C.1, we conclude that the distribution of $w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}$ does not have atoms. In particular, we obtain ${\mathbb{P}}(w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}} = 0) = 0$ .

If the conditions (i) and (ii) hold, but ${{\boldsymbol{{X}}}}_0 \stackrel{{\mathrm{a.s.}}}{=} {\boldsymbol{0}}$ does not necessarily hold, then we apply Lemma 5 with $T = 0$ , and we obtain that ${{\boldsymbol{{X}}}}_t \stackrel{{\mathcal D}}{=} {{\boldsymbol{{X}}}}_t^{(1)} + {{\boldsymbol{{X}}}}_t^{(2,0)}$ for each $t \in \mathbb{R}_+$ , where $({{\boldsymbol{{X}}}}_s^{(1)})_{s\in\mathbb{R}_+}$ and $({{\boldsymbol{{X}}}}_s^{(2,0)})_{s\in\mathbb{R}_+}$ are independent multi-type CBI processes with ${{\boldsymbol{{X}}}}_0^{(1)} \stackrel{{\mathrm{a.s.}}}{=} {\boldsymbol{0}}$ , ${{\boldsymbol{{X}}}}_0^{(2,0)} \stackrel{{\mathcal D}}{=} {{\boldsymbol{{X}}}}_0$ , and with parameters $(d, {{\boldsymbol{{c}}}}, {\boldsymbol{\beta}}, {{\boldsymbol{{B}}}}, \nu, \boldsymbol{\mu})$ and $(d, {{\boldsymbol{{c}}}}, {\boldsymbol{0}}, {{\boldsymbol{{B}}}}, 0, \boldsymbol{\mu})$ , respectively. Then, for each $t \in \mathbb{R}_+$ , we have

\begin{equation*}\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle \stackrel{{\mathcal D}}{=} \mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t^{(1)}\rangle + \mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t^{(2,0)}\rangle.\end{equation*}

By (3.2), we obtain

\begin{equation*}w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0} \stackrel{{\mathcal D}}{=} w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}^{(1)} + w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,0)}}^{(2,0)},\end{equation*}

where $w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}^{(1)}$ and $w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,0)}}^{(2,0)}$ denote the almost sure limits of $\mathrm{e}^{-\lambda s} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_s^{(1)}\rangle$ and of $\mathrm{e}^{-\lambda s} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_s^{(2,0)}\rangle$ , respectively, as $s \to \infty$ . The independence of $({{\boldsymbol{{X}}}}_s^{(1)})_{s\in\mathbb{R}_+}$ and $({{\boldsymbol{{X}}}}_s^{(2,0)})_{s\in\mathbb{R}_+}$ implies the independence of $w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}^{(1)}$ and $w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,0)}}^{(2,0)}$ . We have already shown that $w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}^{(1)} \stackrel{{\mathcal D}}{=} w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}$ does not have atoms, which implies that $w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0}$ does not have atoms, since for each $z \in \mathbb{C}$ , we have

\begin{align*} {\mathbb{P}}(w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0} = z) & = {\mathbb{P}}\Bigl(w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}^{(1)} = z - w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,0)}}^{(2,0)}\Bigr) = {\mathbb{E}}\Bigl({\mathbb{P}}\Bigl(w_{{{\boldsymbol{{v}}}},{\boldsymbol{0}}}^{(1)} = z - w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,0)}}^{(2,0)} \,\Big|\, w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0^{(2,0)}}^{(2,0)}\Bigr)\Bigr)\\ & = {\mathbb{E}}(0) = 0 . \end{align*}

In particular, we obtain ${\mathbb{P}}(w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0} = 0) = 0$ .

If the condition (ii) does not hold, then, as in the proof that Part (ii) $\Longrightarrow$ Part (iii) in Lemma B.1, we obtain that in the representation (B.1) of $\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle$ , the terms $Z_t^{(2)}$ , $Z_t^{(3,4)}$ , and $Z_t^{(5)}$ are 0 almost surely, so $\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle = \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle + \langle{{\boldsymbol{{v}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle \int_0^t \mathrm{e}^{- \lambda u} \, \mathrm{d} u$ for all $t \in \mathbb{R}_+$ almost surely, and hence, taking the limit as $t\to\infty$ , we have $w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0} = \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle + \lambda^{-1} \langle{{\boldsymbol{{v}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle$ almost surely.

If $\lambda = s({\widetilde{{{\boldsymbol{{B}}}}}})$ , ${{\boldsymbol{{v}}}} = {{\boldsymbol{{u}}}}$ , and the conditions (i) and (ii) hold, then we have already derived ${\mathbb{P}}(w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} = 0) = 0$ .

If $\lambda = s({\widetilde{{{\boldsymbol{{B}}}}}})$ , ${{\boldsymbol{{v}}}} = {{\boldsymbol{{u}}}}$ , and the condition (i) holds but the condition (ii) does not hold, then we have already derived

\begin{equation*}{\mathbb{P}}(w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} = 0) = {\mathbb{P}}(\langle{{\boldsymbol{{u}}}}, {{\boldsymbol{{X}}}}_0\rangle + s({\widetilde{{{\boldsymbol{{B}}}}}})^{-1} \langle{{\boldsymbol{{u}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle = 0),\end{equation*}

and this probability is 0, since ${{\boldsymbol{{u}}}} \in \mathbb{R}^d_{++}$ , ${\mathbb{P}}({{\boldsymbol{{X}}}}_0 \in \mathbb{R}^d_+) = 1$ , $s({\widetilde{{{\boldsymbol{{B}}}}}}) \in \mathbb{R}_{++}$ , and $\widetilde{{\boldsymbol{\beta}}} \in \mathbb{R}^d_+\setminus \{{\boldsymbol{0}}\}$ , yielding that $\langle{{\boldsymbol{{u}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle > 0$ .

If $\lambda = s({\widetilde{{{\boldsymbol{{B}}}}}})$ , ${{\boldsymbol{{v}}}} = {{\boldsymbol{{u}}}}$ , and the conditions (i) and (ii) do not hold, then we have already derived

\begin{equation*}{\mathbb{P}}(w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} = 0) = {\mathbb{P}}(\langle{{\boldsymbol{{u}}}}, {{\boldsymbol{{X}}}}_0\rangle + s({\widetilde{{{\boldsymbol{{B}}}}}})^{-1} \langle{{\boldsymbol{{u}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle = 0) = {\mathbb{P}}(\langle{{\boldsymbol{{u}}}}, {{\boldsymbol{{X}}}}_0\rangle = 0),\end{equation*}

and this equals ${\mathbb{P}}({{\boldsymbol{{X}}}}_0 = {\boldsymbol{0}})$ , since ${{\boldsymbol{{u}}}} \in \mathbb{R}^d_{++}$ and ${\mathbb{P}}({{\boldsymbol{{X}}}}_0 \in \mathbb{R}^d_+) = 1$ .

Proof of Lemma 3.1. Note that $w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \stackrel{{\mathrm{a.s.}}}{=} 0$ if and only if ${\mathbb{E}}(w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}) = 0$ . By (3.2), we have $\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t} \langle{{\boldsymbol{{u}}}}, {{\boldsymbol{{X}}}}_t\rangle \stackrel{L_1}{\longrightarrow} w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}$ as $t \to \infty$ . By (2.2), we obtain ${\mathbb{E}}({{\boldsymbol{{X}}}}_t) = \mathrm{e}^{t{\widetilde{{{\boldsymbol{{B}}}}}}} {\mathbb{E}}({{\boldsymbol{{X}}}}_0) + \int_0^t \mathrm{e}^{u{\widetilde{{{\boldsymbol{{B}}}}}}} \widetilde{{\boldsymbol{\beta}}} \, \mathrm{d} u$ , $t \in \mathbb{R}_+$ , so

\begin{align*} {\mathbb{E}}(\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t} \langle{{\boldsymbol{{u}}}}, {{\boldsymbol{{X}}}}_t\rangle) & = \langle{{\boldsymbol{{u}}}}, {\mathbb{E}}({{\boldsymbol{{X}}}}_0) \rangle + \langle{{\boldsymbol{{u}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle \int_0^t \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})(t-u)} \, \mathrm{d} u \\ & \to \langle{{\boldsymbol{{u}}}}, {\mathbb{E}}({{\boldsymbol{{X}}}}_0) \rangle + \frac{\langle{{\boldsymbol{{u}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle}{s({\widetilde{{{\boldsymbol{{B}}}}}})} = {\mathbb{E}}(w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}) \end{align*}

as $t \to \infty$ , where ${{\boldsymbol{{u}}}} \in \mathbb{R}_{++}^d$ and $s({\widetilde{{{\boldsymbol{{B}}}}}}) > 0$ . Thus ${\mathbb{E}}(w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}) = 0$ if and only if ${{\boldsymbol{{X}}}}_0 \stackrel{{\mathrm{a.s.}}}{=} {\boldsymbol{0}}$ and $\widetilde{{\boldsymbol{\beta}}} = {\boldsymbol{0}}$ .

Proof of Part (i) of Theorem 3.3. By Theorem 3.3 in Barczy et al. [Reference Barczy, Palau and Pap8], we have $\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t} {{\boldsymbol{{X}}}}_t \stackrel{{\mathrm{a.s.}}}{\longrightarrow} w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} {\widetilde{{{\boldsymbol{{u}}}}}}$ as $t \to \infty$ ; hence

\begin{equation*}{\mathbb{1}}_{\{{{\boldsymbol{{X}}}}_t\ne{\boldsymbol{0}}\}} = {\mathbb{1}}_{\{\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t}{{\boldsymbol{{X}}}}_t\ne{\boldsymbol{0}}\}} \to 1\end{equation*}

as $t \to \infty$ on the event $\{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} > 0\}$ , since ${\widetilde{{{\boldsymbol{{u}}}}}} \in \mathbb{R}_{++}^d$ . By (3.2), we have

\begin{equation*}\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t} \langle{{\boldsymbol{{u}}}}, {{\boldsymbol{{X}}}}_t\rangle \stackrel{{\mathrm{a.s.}}}{\longrightarrow} w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}, \qquad \mathrm{e}^{-\lambda t} \langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle \stackrel{{\mathrm{a.s.}}}{\longrightarrow} w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0}\end{equation*}

as $t \to \infty$ . Using that $\langle{{\boldsymbol{{u}}}}, {{\boldsymbol{{X}}}}_t\rangle \ne 0$ if and only if ${{\boldsymbol{{X}}}}_t \ne {\boldsymbol{0}}$ , we have

\begin{align*} &{\mathbb{1}}_{\{{{\boldsymbol{{X}}}}_t\ne{\boldsymbol{0}}\}} \frac{1}{\langle {{\boldsymbol{{u}}}}, {{\boldsymbol{{X}}}}_t \rangle^{{\operatorname{Re}}(\lambda)/s({\widetilde{{{\boldsymbol{{B}}}}}})}} \begin{pmatrix} \cos({\operatorname{Im}}(\lambda)t)\;\;\;\;\; & \sin({\operatorname{Im}}(\lambda)t) \\ - \sin({\operatorname{Im}}(\lambda)t)\;\;\;\;\; & \cos({\operatorname{Im}}(\lambda)t) \end{pmatrix} \begin{pmatrix} {\operatorname{Re}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \\ {\operatorname{Im}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \end{pmatrix} \\ &= \frac{{\mathbb{1}}_{\{{{\boldsymbol{{X}}}}_t\ne{\boldsymbol{0}}\}}} {(\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t}\langle{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_t\rangle)^{{\operatorname{Re}}(\lambda)/s({\widetilde{{{\boldsymbol{{B}}}}}})}} \begin{pmatrix} {\operatorname{Re}}(\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle) \\ {\operatorname{Im}}(\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_t\rangle) \end{pmatrix} \to \frac{1}{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}^{{\operatorname{Re}}(\lambda)/s({\widetilde{{{\boldsymbol{{B}}}}}})}} \begin{pmatrix} {\operatorname{Re}}(w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0}) \\ {\operatorname{Im}}(w_{{{\boldsymbol{{v}}}},{{\boldsymbol{{X}}}}_0}) \end{pmatrix} \end{align*}

as $t \to \infty$ on the event $\{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} > 0\}$ , as desired.

Proof of Part (iii) of Theorem 3.3. First, note that the moment condition (3.3) implies the moment condition (3.1) with $\lambda=s({\widetilde{{{\boldsymbol{{B}}}}}})$ , so, by Lemma 3.1, $w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \stackrel{{\mathrm{a.s.}}}{=} 0$ if and only if $({{\boldsymbol{{X}}}}_t)_{t\in\mathbb{R}_+}$ is trivial. For each $t \in \mathbb{R}_+$ , we have the decomposition

\begin{align*} & {\mathbb{1}}_{\{{{\boldsymbol{{X}}}}_t\ne{\boldsymbol{0}}\}} \frac{1}{\sqrt{\langle {{\boldsymbol{{u}}}}, {{\boldsymbol{{X}}}}_t \rangle}} \begin{pmatrix} {\operatorname{Re}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \\ {\operatorname{Im}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \end{pmatrix} \\ &\qquad = {\mathbb{1}}_{\{{{\boldsymbol{{X}}}}_t\ne{\boldsymbol{0}}\}} \frac{\sqrt{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}}} {\sqrt{\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t} \langle {{\boldsymbol{{u}}}}, {{\boldsymbol{{X}}}}_t \rangle}} \frac{\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t/2}}{\sqrt{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}}} \begin{pmatrix} {\operatorname{Re}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \\ {\operatorname{Im}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \end{pmatrix} \end{align*}

on the event $\{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} > 0\}$ . As we have seen in the proof of Part (i) of Theorem 3.3, we have ${\mathbb{1}}_{\{{{\boldsymbol{{X}}}}_t\ne{\boldsymbol{0}}\}} \to 1$ as $t \to \infty$ on the event $\{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} > 0\}$ , and

\begin{equation*}\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t} \langle{{\boldsymbol{{u}}}}, {{\boldsymbol{{X}}}}_t\rangle \stackrel{{\mathrm{a.s.}}}{\longrightarrow} w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}\end{equation*}

as $t\to\infty$ . In the case ${\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}} = {\boldsymbol{0}}$ , (3.6) yields

\begin{equation*} \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t/2} \begin{pmatrix} {\operatorname{Re}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \\ {\operatorname{Im}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \end{pmatrix} \stackrel{\mathbb{P}}{\longrightarrow} {\boldsymbol{0}} \qquad \text{as $t \to \infty$;} \end{equation*}

hence, using the above decomposition, by Slutsky’s lemma we obtain

\begin{equation*} {\mathbb{1}}_{\{{{\boldsymbol{{X}}}}_t\ne{\boldsymbol{0}}\}} \frac{1}{\sqrt{\langle {{\boldsymbol{{u}}}}, {{\boldsymbol{{X}}}}_t \rangle}} \begin{pmatrix} {\operatorname{Re}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \\ {\operatorname{Im}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \end{pmatrix} \stackrel{{\mathcal D}_{\{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0>0}\}}}{\longrightarrow} {\boldsymbol{0}} \qquad \text{as $t \to \infty$.} \end{equation*}

In the case ${\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}} \ne {\boldsymbol{0}}$ , as in the proof of (3.4), we may apply Theorem E.1 to obtain

\begin{equation*} \biggl(\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t/2} \begin{pmatrix} {\operatorname{Re}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \\ {\operatorname{Im}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \end{pmatrix}, \sqrt{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}}\biggr) \stackrel{{\mathcal D}}{\longrightarrow} \big((w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} {\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}})^{1/2} {{\boldsymbol{{N}}}}, \sqrt{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}}\big) \end{equation*}

as $t \to \infty$ , where ${{\boldsymbol{{N}}}}$ is a 2-dimensional random vector with ${{\boldsymbol{{N}}}} \stackrel{{\mathcal D}}{=} {\mathcal N}_2({\boldsymbol{0}}, {{\boldsymbol{{I}}}}_2)$ independent of $w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} {\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}}$ , and hence independent of $w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}$ because ${\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}} \ne {\boldsymbol{0}}$ and ${\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}}$ is deterministic. Applying the continuous mapping theorem, we get

\begin{equation*} \frac{\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t/2}}{\sqrt{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}}} \begin{pmatrix} {\operatorname{Re}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \\ {\operatorname{Im}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \end{pmatrix} \stackrel{{\mathcal D}_{\{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0>0}\}}}{\longrightarrow} {\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}}^{1/2} {{\boldsymbol{{N}}}} \qquad \text{as $t \to \infty$.} \end{equation*}

Hence, again using the above decomposition, by Slutsky’s lemma and (3.2), we have

\begin{align*} {\mathbb{1}}_{\{{{\boldsymbol{{X}}}}_t\ne{\boldsymbol{0}}\}} \frac{1}{\sqrt{\langle {{\boldsymbol{{u}}}}, {{\boldsymbol{{X}}}}_t \rangle}} \begin{pmatrix} {\operatorname{Re}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \\ {\operatorname{Im}}(\langle {{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t \rangle) \end{pmatrix} &\stackrel{{\mathcal D}_{\{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0>0}\}}}{\longrightarrow} {\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}}^{1/2} {{\boldsymbol{{N}}}} \qquad \text{as $t \to \infty$,} \end{align*}

where ${\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}}^{1/2} {{\boldsymbol{{N}}}} \stackrel{{\mathcal D}}{=} {\mathcal N}_2({\boldsymbol{0}}, {\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}})$ , as desired.

Proof of Part (ii) of Theorem 3.3. This is similar to the proof of Part (iii) of Theorem 3.3. For details, see our ArXiv preprint [Reference Barczy, Palau and Pap9].

Proof of Proposition 3.3. Theorem 3.3 in Barczy et al. [Reference Barczy, Palau and Pap8] yields that

\begin{equation*} \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t} \langle{{\boldsymbol{{e}}}}_i, {{\boldsymbol{{X}}}}_t\rangle \stackrel{{\mathrm{a.s.}}}{\longrightarrow} w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \langle{{\boldsymbol{{e}}}}_i, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \qquad \text{and} \qquad \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t} \langle{{\boldsymbol{{e}}}}_j, {{\boldsymbol{{X}}}}_t\rangle \stackrel{{\mathrm{a.s.}}}{\longrightarrow} w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} \langle{{\boldsymbol{{e}}}}_j, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \end{equation*}

as $t \to \infty$ . Consequently, since ${\widetilde{{{\boldsymbol{{u}}}}}} \in \mathbb{R}_{++}$ , we have

\begin{equation*}{\mathbb{1}}_{\{\langle{{\boldsymbol{{e}}}}_j,{{\boldsymbol{{X}}}}_t\rangle\ne0\}} = {\mathbb{1}}_{\{\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t}\langle{{\boldsymbol{{e}}}}_j,{{\boldsymbol{{X}}}}_t\rangle\ne 0\}} \to 1\end{equation*}

as $t \to \infty$ on the event $\{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} > 0\}$ , and hence

\begin{align*} {\mathbb{1}}_{\{\langle{{\boldsymbol{{e}}}}_j,{{\boldsymbol{{X}}}}_t\rangle\ne0\}} \frac{\langle{{\boldsymbol{{e}}}}_i,{{\boldsymbol{{X}}}}_t\rangle}{\langle{{\boldsymbol{{e}}}}_j,{{\boldsymbol{{X}}}}_t\rangle} = {\mathbb{1}}_{\{\langle{{\boldsymbol{{e}}}}_j,{{\boldsymbol{{X}}}}_t\rangle\ne0\}} \frac{\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t}\langle{{\boldsymbol{{e}}}}_i,{{\boldsymbol{{X}}}}_t\rangle} {\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t}\langle{{\boldsymbol{{e}}}}_j,{{\boldsymbol{{X}}}}_t\rangle} \to \frac{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}\langle{{\boldsymbol{{e}}}}_i,{\widetilde{{{\boldsymbol{{u}}}}}}\rangle} {w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}\langle{{\boldsymbol{{e}}}}_j,{\widetilde{{{\boldsymbol{{u}}}}}}\rangle} = \frac{\langle{{\boldsymbol{{e}}}}_i,{\widetilde{{{\boldsymbol{{u}}}}}}\rangle}{\langle{{\boldsymbol{{e}}}}_j,{\widetilde{{{\boldsymbol{{u}}}}}}\rangle} \end{align*}

as $t \to \infty$ on the event $\{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} > 0\}$ ; thus we obtain the first convergence. In a similar way, ${\mathbb{1}}_{\{{{\boldsymbol{{X}}}}_t\ne{\boldsymbol{0}}\}} \to 1$ as $t \to \infty$ on the event $\{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} > 0\}$ , so

\begin{align*} {\mathbb{1}}_{\{{{\boldsymbol{{X}}}}_t\ne{\boldsymbol{0}}\}} \frac{\langle{{\boldsymbol{{e}}}}_i,{{\boldsymbol{{X}}}}_t\rangle}{\sum_{k=1}^d\langle{{\boldsymbol{{e}}}}_k,{{\boldsymbol{{X}}}}_t\rangle} & = {\mathbb{1}}_{\{{{\boldsymbol{{X}}}}_t\ne{\boldsymbol{0}}\}} \frac{\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t}\langle{{\boldsymbol{{e}}}}_i,{{\boldsymbol{{X}}}}_t\rangle} {\sum_{k=1}^d\mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t}\langle{{\boldsymbol{{e}}}}_k,{{\boldsymbol{{X}}}}_t\rangle} \\ &\, \to \frac{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}\langle{{\boldsymbol{{e}}}}_i,{\widetilde{{{\boldsymbol{{u}}}}}}\rangle} {\sum_{k=1}^dw_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0}\langle{{\boldsymbol{{e}}}}_k,{\widetilde{{{\boldsymbol{{u}}}}}}\rangle} = \langle{{\boldsymbol{{e}}}}_i, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle \end{align*}

as $t \to \infty$ on the event $\{w_{{{\boldsymbol{{u}}}},{{\boldsymbol{{X}}}}_0} > 0\}$ , since the sum of the coordinates of ${\widetilde{{{\boldsymbol{{u}}}}}}$ is 1. Thus we obtain the second convergence.

Appendix A. A decomposition of multi-type CBI processes

The following useful decomposition of a multi-type CBI process as an independent sum of a CBI process starting from ${\boldsymbol{0}}$ and a CB process is derived in Barczy et al. [Reference Barczy, Palau and Pap8, Lemma A.1].

Appendix B. On deterministic projections of multi-type CBI processes

1. (i) There exists $t \in \mathbb{R}_{++}$ such that $\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle$ is deterministic.

2. (ii) One of the following two conditions holds:

3. (a) $({{\boldsymbol{{X}}}}_t)_{t\in\mathbb{R}_+}$ is a trivial process (see Definition 2.4).

4. (b) $\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle$ is deterministic, $\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle c_\ell = 0$ and $\mu_\ell(\{{{\boldsymbol{{z}}}} \in {\mathcal U}_d : \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle \ne 0\}) = 0$ for every $\ell \in \{1, \ldots, d\}$ , and $\nu(\{{{\boldsymbol{{r}}}} \in {\mathcal U}_d : \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle \ne 0\}) = 0$ .

5. (iii) For each $t \in \mathbb{R}_+$ , $\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle$ is deterministic.

If $(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle)_{t\in\mathbb{R}_+}$ is deterministic, then $\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle \stackrel{{\mathrm{a.s.}}}{=} \mathrm{e}^{\lambda t} \langle{{\boldsymbol{{v}}}}, {\mathbb{E}}({{\boldsymbol{{X}}}}_0)\rangle + \langle{{\boldsymbol{{v}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle \int_0^t \mathrm{e}^{\lambda u} \, \mathrm{d} u$ , $t \in \mathbb{R}_+$ .

Proof. Proof that (i) $\Longrightarrow$ (ii): We have the representation

(B.1) \begin{equation} \mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle = Z_t^{(0)} + Z_t^{(1)} + Z_t^{(2)} + Z_t^{(3,4)} + Z_t^{(5)} \end{equation}

with

\begin{align*} Z_t^{(0)} &\,:\!=\, \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle , \\ Z_t^{(1)} &\,:\!=\, \langle{{\boldsymbol{{v}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle \int_0^t \mathrm{e}^{-\lambda u} \, \mathrm{d} u , \\ Z_t^{(2)} &\,:\!=\, \sum_{\ell=1}^d \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle \int_0^t \mathrm{e}^{-\lambda u} \sqrt{2 c_\ell X_{u,\ell}} \, \mathrm{d} W_{u,\ell} , \\ Z_t^{(3,4)} &\,:\!=\, \sum_{\ell=1}^d \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, \widetilde{N}_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) , \\ Z_t^{(5)} &\,:\!=\, \int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle \, \widetilde{M}(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{r}}}}); \end{align*}

see Barczy et al. [Reference Barczy, Li and Pap7, Lemma 4.1] or Barczy et al. [Reference Barczy, Palau and Pap8, Lemma 2.7]. Note that under the moment condition (3.8), $(Z_t^{(2)})_{t\in\mathbb{R}_+}$ , $(Z_t^{(3,4)})_{t\in\mathbb{R}_+}$ and $(Z_t^{(5)})_{t\in\mathbb{R}_+}$ are square-integrable martingales with initial values 0; hence ${\mathbb{E}}(Z_t^{(2)}) = {\mathbb{E}}(Z_t^{(3,4)}) = {\mathbb{E}}(Z_t^{(5)}) = 0$ . Since $\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle$ and $Z_t^{(1)}$ are deterministic, we obtain

\begin{equation*}\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle = {\mathbb{E}}(\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle) = {\mathbb{E}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle) + Z_t^{(1)}.\end{equation*}

Hence, by the representation (B.1), we get

\begin{equation*}0 = \mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle - {\mathbb{E}}(\mathrm{e}^{-\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle) = \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle - {\mathbb{E}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle) + \sum_{j=2}^5 Z_t^{(j)}\end{equation*}

almost surely. Consequently,

\begin{equation*} {\mathbb{E}}(|\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle - {\mathbb{E}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle) + Z_t^{(2)} + Z_t^{(3,4)} + Z_t^{(5)} |^2) = 0. \end{equation*}

By the independence of ${{\boldsymbol{{X}}}}_0$ , $(W_{u,1})_{u\geqslant 0}$ , …, $(W_{u,d})_{u\geqslant 0}$ , $N_1$ , …, $N_d$ , and M, the random variables $\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle - {\mathbb{E}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle)$ , $Z_t^{(2)}$ , $Z_t^{(3,4)}$ , and $Z_t^{(5)}$ are conditionally independent with respect to $({{\boldsymbol{{X}}}}_u)_{u\in[0,t]}$ ; thus, by conditioning on $({{\boldsymbol{{X}}}}_u)_{u\in[0,t]}$ , we have

\begin{align*} 0 &= {\mathbb{E}}\biggl(\biggl|\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle - {\mathbb{E}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle) + Z_t^{(2)} + Z_t^{(3,4)} + Z_t^{(5)} \biggr|^2\biggr) \\ &= {\mathbb{E}}({\mathbb{E}}(|\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle - {\mathbb{E}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle)|^2 \,|\, ({{\boldsymbol{{X}}}}_u)_{u\in[0,t]})) + {\mathbb{E}}({\mathbb{E}}(|Z_t^{(2)}|^2 \,|\, ({{\boldsymbol{{X}}}}_u)_{u\in[0,t]})) \\ &\phantom{=\;} + {\mathbb{E}}({\mathbb{E}}(|Z_t^{(3,4)}|^2 \,|\, ({{\boldsymbol{{X}}}}_u)_{u\in[0,t]})) + {\mathbb{E}}({\mathbb{E}}(|Z_t^{(5)}|^2 \,|\, ({{\boldsymbol{{X}}}}_u)_{u\in[0,t]})) \\ &= {\mathbb{E}}(|\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle - {\mathbb{E}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle)|^2) + {\mathbb{E}}(|Z_t^{(2)}|^2) + {\mathbb{E}}(|Z_t^{(3,4)}|^2) + {\mathbb{E}}(|Z_t^{(5)}|^2), \end{align*}

where we also used that $(Z_s^{(2)})_{s\in[0,t]}$ , $(Z_s^{(3,4)})_{s\in[0,t]}$ , and $(Z_s^{(5)})_{s\in[0,t]}$ are square-integrable martingales with initial values 0 conditionally on $({{\boldsymbol{{X}}}}_u)_{u\in[0,t]}$ . Consequently, ${\mathbb{E}}(|\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle - {\mathbb{E}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle)|^2 ) = 0$ and ${\mathbb{E}}(|Z_t^{(2)}|^2) = {\mathbb{E}}(|Z_t^{(3,4)}|^2) = {\mathbb{E}}(|Z_t^{(5)}|^2) = 0$ . One can easily derive

\begin{equation*} {\mathbb{E}}(|Z_t^{(2)}|^2) = 2 \sum_{\ell=1}^d |\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle|^2 c_\ell \int_0^t \mathrm{e}^{-2{\operatorname{Re}}(\lambda)u} {\mathbb{E}}(X_{u,\ell}) \, \mathrm{d} u , \end{equation*}

so we conclude

\begin{equation*}|\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle|^2 c_\ell \int_0^t \mathrm{e}^{-2{\operatorname{Re}}(\lambda)u} {\mathbb{E}}(X_{u,\ell}) \, \mathrm{d} u = 0, \qquad \ell \in \{1, \ldots, d\}.\end{equation*}

Consequently, for each $\ell \in \{1, \ldots, d\}$ , we have $|\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle|^2 c_\ell = 0$ or $\int_0^t \mathrm{e}^{-2{\operatorname{Re}}(\lambda)u} {\mathbb{E}}(X_{u,\ell}) \, \mathrm{d} u = 0$ . In the first case we obtain $\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle c_\ell = 0$ , which is in (ii)/(b). In the second case, using Lemma 2.1 and $\mathrm{e}^{-2{\operatorname{Re}}(\lambda)u} \in \mathbb{R}_{++}$ for all $u \in \mathbb{R}_+$ , we conclude (ii)/(a).

Since ${\mathbb{E}}(|Z_t^{(3,4)}|^2) = 0$ , we have

\begin{equation*} {\mathbb{E}}(|Z_t^{(3,4)}|^2) = \sum_{\ell=1}^d \int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{-2{\operatorname{Re}}(\lambda)u} |\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle|^2 {\mathbb{E}}(X_{u,\ell}) \, \mathrm{d} u \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) = 0 . \end{equation*}

Using $\mathrm{e}^{-2{\operatorname{Re}}(\lambda)u} \in \mathbb{R}_{++}$ for all $u \in \mathbb{R}_+$ , we conclude

\begin{equation*} \sum_{\ell=1}^d \int_0^t \int_{{\mathcal U}_d} {\mathbb{1}}_{\{\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle\ne0\}} {\mathbb{E}}(X_{u,\ell}) \, \mathrm{d} u \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) = 0. \end{equation*}

Then, using the non-negativity of the integrands, we obtain

\begin{equation*} \int_0^t \int_{{\mathcal U}_d} {\mathbb{1}}_{\{\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle\ne0\}} {\mathbb{E}}(X_{u,\ell}) \, \mathrm{d} u \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) = 0 , \qquad \ell \in \{1, \ldots, d\} . \end{equation*}

By Lemma 2.1, for each $\ell \in \{1, \ldots, d\}$ , we have either (ii)/(a), or ${\mathbb{E}}(X_{u,\ell}) = {{\boldsymbol{{e}}}}_\ell^\top {\mathbb{E}}({{\boldsymbol{{X}}}}_u) \in \mathbb{R}_{++}$ for all $u \in \mathbb{R}_{++}$ . In the second case, we conclude

\begin{equation*} \int_0^t \int_{{\mathcal U}_d} {\mathbb{1}}_{\{\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle\ne0\}} \, \mathrm{d} u \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) = t \mu_\ell(\{{{\boldsymbol{{z}}}} \in {\mathcal U}_d : \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle \ne 0\}) = 0 , \end{equation*}

and hence $\mu_\ell(\{{{\boldsymbol{{z}}}} \in {\mathcal U}_d : \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle \ne 0\}) = 0$ , which is in (ii)/(b).

Since ${\mathbb{E}}(|Z_t^{(5)}|^2) = 0$ , we have $Z_t^{(5)} = 0$ almost surely. Hence the random variable $\int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle \, M(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{r}}}})$ is deterministic, since $\int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle \, \mathrm{d} u \, \nu(\mathrm{d}{{\boldsymbol{{r}}}})$ is deterministic. We have

\begin{equation*}Z_s^{(5)} = {\mathbb{E}}(Z_t^{(5)} \,|\, {\mathcal F}_s^{Z^{(5)}}) = {\mathbb{E}}( 0 \,|\, {\mathcal F}_s^{Z^{(5)}}) = 0\end{equation*}

for all $s \in [0,t]$ almost surely, where

\begin{equation*}{\mathcal F}_s^{Z^{(5)}} \,:\!=\,\sigma(Z^{(5)}_u : u\in[0,s]),\end{equation*}

since $(Z_s^{(5)})_{s\in\mathbb{R}_+}$ is a martingale. Thus ${\mathbb{P}}(A_t^{(M)}) = 1$ , where $A_t^{(M)}$ is the event such that the Poisson random measure M has no point in the set $H_t$ , where

\begin{equation*} H_t \,:\!=\, \{(u, {{\boldsymbol{{r}}}}) \in (0, t] \times {\mathcal U}_d \;:\; \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle \ne 0\} = \{(u, {{\boldsymbol{{r}}}}) \in (0, t] \times {\mathcal U}_d \;:\; {\mathbb{1}}_{\{\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle\ne0\}} \ne 0\} , \end{equation*}

since $\mathrm{e}^{-\lambda u} \ne 0$ for all $u \in \mathbb{R}_+$ . The number of the points of M in the set $H_t$ has a Poisson distribution with parameter

\begin{equation*}\lambda_t \,:\!=\, \int_0^t \int_{{\mathcal U}_d} {\mathbb{1}}_{\{\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle\ne0\}} \, \mathrm{d} u \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}).\end{equation*}

We have $1 = {\mathbb{P}}(A_t^{(M)}) = \mathrm{e}^{-\lambda_t}$ , yielding

\begin{equation*} \lambda_t = \int_0^t \int_{{\mathcal U}_d} {\mathbb{1}}_{\{\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle\ne0\}} \, \mathrm{d} u \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) = t \nu(\{{{\boldsymbol{{r}}}} \in {\mathcal U}_d : \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle \ne 0\}) = 0 , \end{equation*}

and hence $\nu(\{{{\boldsymbol{{r}}}} \in {\mathcal U}_d : \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle \ne 0\}) = 0$ , which is in (ii)/(b).

Proof that (ii) $\Longrightarrow$ (iii): If (ii)/(a) holds, then $\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle \stackrel{{\mathrm{a.s.}}}{=} 0$ for all $t \in \mathbb{R}_+$ . If (ii)/(b) holds, then we use again the representation (B.1) of $\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle$ . We have $\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle = {\mathbb{E}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle) = \langle{{\boldsymbol{{v}}}}, {\mathbb{E}}({{\boldsymbol{{X}}}}_0)\rangle$ , since $\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle$ is deterministic. For each $t \in \mathbb{R}_+$ , we have

\begin{equation*} Z_t^{(2)} = \sqrt{2} \sum_{\ell=1}^d \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle \sqrt{c_\ell} \int_0^t \mathrm{e}^{-\lambda u} \sqrt{X_{u,\ell}} \, \mathrm{d} W_{u,\ell} = 0 , \end{equation*}

since $\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{e}}}}_\ell\rangle c_\ell = 0$ for every $\ell \in \{1, \ldots, d\}$ .

Further, for each $t \in \mathbb{R}_+$ and $n \in \mathbb{N}$ , using the notation

\begin{align*} f(u, {{\boldsymbol{{z}}}}, w) \,:\!=\, \sum_{\ell=1}^d \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} = \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle \sum_{\ell=1}^d {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \end{align*}

for $u \in (0, t]$ , ${{\boldsymbol{{z}}}} \in {\mathcal U}_d$ , and $w \in {\mathcal U}_1$ , we have

\begin{align*} &\int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} {\mathbb{1}}_{\{|f(u,{{\boldsymbol{{z}}}},w)|<n\}} {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|>1/n\}} {\mathbb{1}}_{\{w<n\}} f(u, {{\boldsymbol{{z}}}}, w) \, \widetilde{N}_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) \\ &= \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} {\mathbb{1}}_{\{|f(u,{{\boldsymbol{{z}}}},w)|<n\}} {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|>1/n\}} {\mathbb{1}}_{\{w<n\}} f(u, {{\boldsymbol{{z}}}}, w) \, N_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) \\ &\quad - \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} {\mathbb{1}}_{\{|f(u,{{\boldsymbol{{z}}}},w)|<n\}} {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|>1/n\}} {\mathbb{1}}_{\{w<n\}} f(u, {{\boldsymbol{{z}}}}, w) \, \mathrm{d} u \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \, \mathrm{d} w = 0 \end{align*}

almost surely, since

\begin{equation*}\int_{{\mathcal U}_d} {\mathbb{1}}_{\{\|{{\boldsymbol{{z}}}}\|>1/n\}} \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) \leqslant n^2 \int_{{\mathcal U}_d} \|{{\boldsymbol{{z}}}}\|^2 \, \mu_\ell(\mathrm{d}{{\boldsymbol{{z}}}}) < \infty\end{equation*}

by Part (vi) of Definition 2.1, (3.8), and

\begin{equation*} ({\mathcal L}_1\otimes \mu_\ell\otimes {\mathcal L}_d)(\{ (u,{{\boldsymbol{{z}}}},w) \in (0,t]\times{\mathcal U}_d\times {\mathcal U}_1 \;:\; f(u, {{\boldsymbol{{z}}}}, w) \ne 0\}) = 0 \end{equation*}

for each $\ell \in \{1, \ldots, d\}$ , where ${\mathcal L}_1$ and ${\mathcal L}_d$ denote the Lebesgue measure on $\mathbb{R}$ and on $\mathbb{R}^d$ , respectively. Letting $n \to \infty$ , by Ikeda and Watanabe [Reference Ikeda and Watanabe16, page 63], we conclude

\begin{equation*} Z_t^{(3,4)} = \sum_{\ell=1}^d \int_0^t \int_{{\mathcal U}_d} \int_{{\mathcal U}_1} \mathrm{e}^{-\lambda u} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{z}}}}\rangle {\mathbb{1}}_{\{w\leqslant X_{u-,\ell}\}} \, \widetilde{N}_\ell(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{z}}}}, \mathrm{d} w) \stackrel{{\mathrm{a.s.}}}{=} 0. \end{equation*}

Finally, for each $t \in \mathbb{R}_+$ , we have

\begin{align*} Z^{(5)}_t = \int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{-\lambda u} \langle {{\boldsymbol{{v}}}},{{\boldsymbol{{r}}}}\rangle \, M(\mathrm{d} u, \mathrm{d}{{\boldsymbol{{r}}}}) - \int_0^t \int_{{\mathcal U}_d} \mathrm{e}^{-\lambda u} \langle {{\boldsymbol{{v}}}},{{\boldsymbol{{r}}}}\rangle \, \mathrm{d} u \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) = 0 \end{align*}

almost surely, since $\int_{{\mathcal U}_d} \|{{\boldsymbol{{r}}}}\| \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) < \infty$ (by Definition 2.1 and (2.1)) and $\nu(\{{{\boldsymbol{{r}}}} \in {\mathcal U}_d : \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle \ne 0\}) = 0$ .

The implication (iii) $\Longrightarrow$ (i) is trivial.

If $(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle)_{t\in\mathbb{R}_+}$ is deterministic, then, by (2.2), for each $t \in \mathbb{R}_+$ , we have

\begin{equation*}\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle = {\mathbb{E}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle) = \langle{{\boldsymbol{{v}}}}, {\mathbb{E}}({{\boldsymbol{{X}}}}_t)\rangle = \mathrm{e}^{\lambda t} \langle{{\boldsymbol{{v}}}}, {\mathbb{E}}({{\boldsymbol{{X}}}}_0)\rangle + \langle{{\boldsymbol{{v}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle \int_0^t \mathrm{e}^{\lambda u} \, \mathrm{d} u\end{equation*}

almost surely.

Appendix C. A stochastic fixed point equation

Under some mild conditions, the solution of a stochastic fixed point equation is atomless; see, e.g., Buraczewski et al. [Reference Buraczewski, Damek and Mikosch11, Proposition 4.3.2].

1. (i) ${\boldsymbol{{A}}}$ is invertible almost surely,

2. (ii) ${\mathbb{P}}({\boldsymbol{{A}}} {{\boldsymbol{{x}}}} + {{\boldsymbol{{C}}}} = {{\boldsymbol{{x}}}}) < 1$ for every ${{\boldsymbol{{x}}}} \in \mathbb{R}^d$ , and

3. (iii) the d-dimensional fixed point equation ${{\boldsymbol{{X}}}} \stackrel{{\mathcal D}}{=} {\boldsymbol{{A}}} {{\boldsymbol{{X}}}} + {{\boldsymbol{{C}}}}$ , where $({\boldsymbol{{A}}}, {{\boldsymbol{{C}}}})$ and ${{\boldsymbol{{X}}}}$ are independent, has a solution ${{\boldsymbol{{X}}}}$ , which is unique in distribution.

Then the distribution of ${{\boldsymbol{{X}}}}$ does not have atoms and is of pure type, i.e., it is either absolutely continuous or singular with respect to Lebesgue measure in $\mathbb{R}^d$ .

For a proof of Corollary C.1, see our ArXiv preprint [Reference Barczy, Palau and Pap9].

Appendix D. On the second moment of projections of multi-type CBI processes

An explicit formula for the second absolute moment of the projection of a multi-type CBI process on the left eigenvectors of its branching mean matrix is presented, together with its asymptotic behavior in the supercritical and irreducible case, in Barczy et al. [Reference Barczy, Palau and Pap8, Proposition B.1].

Proposition D.1. If $({{\boldsymbol{{X}}}}_t)_{t\in\mathbb{R}_+}$ is a multi-type CBI process having parameters $(d, {{\boldsymbol{{c}}}}, {\boldsymbol{\beta}}, {{\boldsymbol{{B}}}}, \nu, \boldsymbol{\mu})$ such that ${\mathbb{E}}(\|{{\boldsymbol{{X}}}}_0\|^2) < \infty$ and the moment condition (3.8) holds, then for each left eigenvector ${{\boldsymbol{{v}}}} \in \mathbb{C}^d$ of ${\widetilde{{{\boldsymbol{{B}}}}}}$ corresponding to an arbitrary eigenvalue $\lambda \in \sigma({\widetilde{{{\boldsymbol{{B}}}}}})$ , we have

\begin{equation*} {\mathbb{E}}(|\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle|^2) = E_{{{\boldsymbol{{v}}}},\lambda}(t) + \sum_{\ell=1}^d C_{{{\boldsymbol{{v}}}},\ell} I_{\lambda,\ell}(t) + I_\lambda(t) \int_{{\mathcal U}_d} |\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle|^2 \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) , \qquad t \in \mathbb{R}_+ , \end{equation*}

where $C_{{{\boldsymbol{{v}}}},\ell}$ , $\ell \in \{1, \ldots, d\}$ , are defined in Theorem 3.1, and

\begin{align*} E_{{{\boldsymbol{{v}}}},\lambda}(t) &\,:\!=\, {\mathbb{E}}\biggl(\biggl|\mathrm{e}^{\lambda t} \langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle + \langle{{\boldsymbol{{v}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle \int_0^t \mathrm{e}^{\lambda(t-u)} \, \mathrm{d} u\biggr|^2\biggr) , \\ I_{\lambda,\ell}(t) &\,:\!=\, \int_0^t \mathrm{e}^{2{\operatorname{Re}}(\lambda)(t-u)} {\mathbb{E}}(X_{u,\ell}) \, \mathrm{d} u , \qquad \ell \in \{1, \ldots, d\} , \\ I_\lambda(t) &\,:\!=\, \int_0^t \mathrm{e}^{2{\operatorname{Re}}(\lambda)(t-u)} \, \mathrm{d} u . \end{align*}

If, in addition, $({{\boldsymbol{{X}}}}_t)_{t\in\mathbb{R}_+}$ is supercritical and irreducible, then we have

\begin{equation*} \lim_{t\to\infty} h(t) {\mathbb{E}}(|\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle|^2) = M_{{\boldsymbol{{v}}}}^{(2)} , \end{equation*}

where

\begin{gather*} h(t) \,:\!=\, \begin{cases} \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t} & \text{if ${\operatorname{Re}}(\lambda) \in \bigl(-\infty, \frac{1}{2} s({\widetilde{{{\boldsymbol{{B}}}}}})\bigr)$,} \\ t^{-1} \mathrm{e}^{-s({\widetilde{{{\boldsymbol{{B}}}}}})t} & \text{if ${\operatorname{Re}}(\lambda) = \frac{1}{2} s({\widetilde{{{\boldsymbol{{B}}}}}})$,} \\ \mathrm{e}^{-2{\operatorname{Re}}(\lambda)t} & \text{if ${\operatorname{Re}}(\lambda) \in \bigl(\frac{1}{2} s({\widetilde{{{\boldsymbol{{B}}}}}}), s({\widetilde{{{\boldsymbol{{B}}}}}})\bigr]$,} \end{cases} \end{gather*}

and $M_{{\boldsymbol{{v}}}}^{(2)}$ is given by

\begin{gather*} \begin{cases} \frac{1}{s^{^{^{^{}}}}\!({\widetilde{{{\boldsymbol{{B}}}}}})-2{\operatorname{Re}}(\lambda)} \bigl(\langle{{\boldsymbol{{u}}}}, {\mathbb{E}}({{\boldsymbol{{X}}}}_0)\rangle + \frac{\langle{{\boldsymbol{{u}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle}{s^{^{^{^{}}}}\!({\widetilde{{{\boldsymbol{{B}}}}}})}\bigr) \sum_{\ell=1}^d C_{{{\boldsymbol{{v}}}},\ell} \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle & \text{if ${\operatorname{Re}}(\lambda) \in \bigl(-\infty, \frac{1}{2} s({\widetilde{{{\boldsymbol{{B}}}}}})\bigr)$,} \\ \bigl(\langle{{\boldsymbol{{u}}}}, {\mathbb{E}}({{\boldsymbol{{X}}}}_0)\rangle + \frac{\langle{{\boldsymbol{{u}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle}{s^{^{^{^{}}}}\!({\widetilde{{{\boldsymbol{{B}}}}}})}\bigr) \sum_{\ell=1}^d C_{{{\boldsymbol{{v}}}},\ell} \langle{{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle & \text{if ${\operatorname{Re}}(\lambda) = \frac{1}{2} s({\widetilde{{{\boldsymbol{{B}}}}}})$,} \\[4pt] {\mathbb{E}}\bigl(\bigl|\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_0\rangle + \frac{\langle{{\boldsymbol{{v}}}}, \widetilde{{\boldsymbol{\beta}}}\rangle}{\lambda}\bigr|^2\bigr) + \frac{1}{2{\operatorname{Re}}(\lambda)} \int_{{\mathcal U}_d} |\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{r}}}}\rangle|^2 \, \nu(\mathrm{d}{{\boldsymbol{{r}}}}) \\ + \sum\limits_{\ell=1}^d C_{{{\boldsymbol{{v}}}},\ell} {{\boldsymbol{{e}}}}_\ell^\top (2 {\operatorname{Re}}(\lambda) {{\boldsymbol{{I}}}}_d - {\widetilde{{{\boldsymbol{{B}}}}}})^{-1} \bigl({\mathbb{E}}({{\boldsymbol{{X}}}}_0) + \frac{\widetilde{{\boldsymbol{\beta}}}}{2{\operatorname{Re}}(\lambda)}\bigr) & \text{if ${\operatorname{Re}}(\lambda) \in \bigl(\frac{1}{2} s({\widetilde{{{\boldsymbol{{B}}}}}}), s({\widetilde{{{\boldsymbol{{B}}}}}})\bigr]$.} \end{cases} \end{gather*}

Based on Proposition D.1, we derive the asymptotic behavior of the variance matrix of the real and imaginary parts of the projection of a multi-type CBI process on certain left eigenvectors of its branching mean matrix $\mathrm{e}^{\widetilde{{{\boldsymbol{{B}}}}}}$ .

Proposition D.2. If $({{\boldsymbol{{X}}}}_t)_{t\in\mathbb{R}_+}$ is a supercritical and irreducible multi-type CBI process with parameters $(d, {{\boldsymbol{{c}}}}, {\boldsymbol{\beta}}, {{\boldsymbol{{B}}}}, \nu, \boldsymbol{\mu})$ such that ${\mathbb{E}}(\|{{\boldsymbol{{X}}}}_0\|^2) < \infty$ and the moment condition (3.8) holds, then for each left eigenvector ${{\boldsymbol{{v}}}} \in \mathbb{C}^d$ of ${\widetilde{{{\boldsymbol{{B}}}}}}$ corresponding to an arbitrary eigenvalue $\lambda \in \sigma({\widetilde{{{\boldsymbol{{B}}}}}})$ with ${\operatorname{Re}}(\lambda) \in \bigl(-\infty, \frac{1}{2} s({\widetilde{{{\boldsymbol{{B}}}}}})\bigr]$ , we have

\begin{equation*} \lim_{t\to\infty} h(t) {\mathbb{E}}\left(\begin{pmatrix} {\operatorname{Re}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle) \\ {\operatorname{Im}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle) \end{pmatrix} \begin{pmatrix} {\operatorname{Re}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle) \\ {\operatorname{Im}}(\langle{{\boldsymbol{{v}}}}, {{\boldsymbol{{X}}}}_t\rangle) \end{pmatrix}^\top\right) = \biggl(\langle{{\boldsymbol{{u}}}}, {\mathbb{E}}({{\boldsymbol{{X}}}}_0)\rangle + \frac{\langle{{\boldsymbol{{u}}}},\widetilde{{\boldsymbol{\beta}}}\rangle}{s({\widetilde{{{\boldsymbol{{B}}}}}})}\biggr) {\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}} , \end{equation*}

where the scaling factor $h \;:\; \mathbb{R}_{++} \to \mathbb{R}_{++}$ and the matrix ${\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}}$ are defined in Proposition D.1 and in Theorem 3.1, respectively.

For a proof of Proposition D.2, see our ArXiv preprint [Reference Barczy, Palau and Pap9].

For a proof of Proposition D.3, see our ArXiv preprint [Reference Barczy, Palau and Pap9].

Remark D.1. Under the conditions of Proposition D.3, if $\lambda \in \sigma({\widetilde{{{\boldsymbol{{B}}}}}})$ with ${\operatorname{Re}}(\lambda) \in \bigl(-\infty, \frac{1}{2} s({\widetilde{{{\boldsymbol{{B}}}}}})\bigr]$ and ${\operatorname{Im}}(\lambda) = 0$ and ${{\boldsymbol{{v}}}} \in \mathbb{R}^d$ is a left eigenvector of ${\widetilde{{{\boldsymbol{{B}}}}}}$ corresponding to the eigenvalue $\lambda$ , then ${\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}}$ is singular. Indeed, in this case, by (3.7) and (3.5), we have

\begin{equation*} {\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}} = \begin{cases} \begin{pmatrix} \frac{1}{s^{^{^{^{}}}}\!({\widetilde{{{\boldsymbol{{B}}}}}})-2\lambda} \sum_{\ell=1}^d \langle {{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle C_{{{\boldsymbol{{v}}}},\ell}\;\;\;\;\;\;\; & 0 \\ 0\;\;\;\;\;\;\; & 0 \\ \end{pmatrix} & \text{if ${\operatorname{Re}}(\lambda) \in \bigl(-\infty, \frac{1}{2} s({\widetilde{{{\boldsymbol{{B}}}}}})\bigr)$,} \\ \begin{pmatrix} \sum_{\ell=1}^d \langle {{\boldsymbol{{e}}}}_\ell, {\widetilde{{{\boldsymbol{{u}}}}}}\rangle C_{{{\boldsymbol{{v}}}},\ell}\;\;\;\;\;\;\; & 0 \\ 0\;\;\;\;\;\;\; & 0 \\ \end{pmatrix} & \text{if ${\operatorname{Re}}(\lambda) = \frac{1}{2} s({\widetilde{{{\boldsymbol{{B}}}}}})$.} \end{cases} \end{equation*}

Note that if ${{\boldsymbol{{v}}}} \in \mathbb{R}^d$ is a left eigenvector of ${\widetilde{{{\boldsymbol{{B}}}}}}$ corresponding to an eigenvalue $\lambda$ of ${\widetilde{{{\boldsymbol{{B}}}}}}$ , then $\lambda \in \mathbb{R}$ necessarily, and hence in the case $\lambda\in(-\infty,\frac{1}{2}s({\widetilde{{{\boldsymbol{{B}}}}}})]$ , we have that ${\boldsymbol{\Sigma}}_{{\boldsymbol{{v}}}}$ is not invertible. However, if $\lambda \in \mathbb{R}$ is an eigenvalue of ${\widetilde{{{\boldsymbol{{B}}}}}}$ and ${{\boldsymbol{{v}}}} \in \mathbb{C}^d$ is a left eigenvector of ${\widetilde{{{\boldsymbol{{B}}}}}}$ corresponding to $\lambda$ , then ${\operatorname{Re}}({{\boldsymbol{{v}}}}) \in \mathbb{R}^d$ and ${\operatorname{Im}}({{\boldsymbol{{v}}}}) \in \mathbb{R}^d$ are also left eigenvectors of ${\widetilde{{{\boldsymbol{{B}}}}}}$ or the zero vector.

Appendix E. A limit theorem for martingales

The next theorem is about the asymptotic behavior of multivariate martingales.

Theorem E.1. (Crimaldi and Pratelli [Reference Crimaldi and Pratelli12, Theorem 2.2]) Let us consider a filtered probability space $\bigl(\Omega, {\mathcal F}, ({\mathcal F}_t)_{t\in\mathbb{R}_+}, {\mathbb{P}}\bigr)$ satisfying the usual conditions. Let $({{\boldsymbol{{M}}}}_t)_{t\in\mathbb{R}_+}$ be a d-dimensional martingale with respect to the filtration $({\mathcal F}_t)_{t\in\mathbb{R}_+}$ such that it has cÀdlÀg sample paths almost surely. Suppose that there exists a function ${{\boldsymbol{{Q}}}} : \mathbb{R}_+ \to \mathbb{R}^{d \times d}$ such that $\lim_{t\to\infty} {{\boldsymbol{{Q}}}}(t) = {\boldsymbol{0}}$ ;

(E.1) \begin{equation} {{\boldsymbol{{Q}}}}(t) [{{\boldsymbol{{M}}}}]_t {{\boldsymbol{{Q}}}}(t)^\top \stackrel{\mathbb{P}}{\longrightarrow} {\boldsymbol{\eta}} \qquad \text{as $t \to \infty$,} \end{equation}

where ${\boldsymbol{\eta}}$ is a $d \times d$ random (necessarily positive semidefinite) matrix and $([{{\boldsymbol{{M}}}}]_t)_{t\in\mathbb{R}_+}$ denotes the quadratic variation (matrix-valued) process of $({{\boldsymbol{{M}}}}_t)_{t\in\mathbb{R}_+}$ ; and

(E.2) \begin{equation} {\mathbb{E}}\biggl(\sup_{u\in[0,t]} \|{{\boldsymbol{{Q}}}}(t) ({{\boldsymbol{{M}}}}_u - {{\boldsymbol{{M}}}}_{u-})\|\biggr) \to 0 \qquad \text{as $t \to \infty$.} \end{equation}

Then, for each $\mathbb{R}^{k\times\ell}$ -valued random matrix ${\boldsymbol{{A}}}$ defined on $(\Omega, {\mathcal F}, {\mathbb{P}})$ with some $k, \ell \in \mathbb{N}$ , we have

\begin{equation*} ({{\boldsymbol{{Q}}}}(t) {{\boldsymbol{{M}}}}_t, {\boldsymbol{{A}}}) \stackrel{{\mathcal D}}{\longrightarrow} ({\boldsymbol{\eta}}^{1/2} {{\boldsymbol{{Z}}}}, {\boldsymbol{{A}}}) \qquad \text{as $t \to \infty$,} \end{equation*}

where ${{\boldsymbol{{Z}}}}$ is a d-dimensional random vector with ${{\boldsymbol{{Z}}}} \stackrel{{\mathcal D}}{=} {\mathcal N}_d({\boldsymbol{0}}, {{\boldsymbol{{I}}}}_d)$ independent of $({\boldsymbol{\eta}}, {\boldsymbol{{A}}})$ .