2.1 Some special models

Several examples of Poisson cluster processes have been studied in the monograph of Daley and Vere Jones (Reference Daley and Vere-Jones2003); see Example 6.3 therein for instance. Here we study the marked adaptation of Example 6.3(a)–(c) of Daley and Vere Jones (Reference Daley and Vere-Jones2003).

2.1.1 Mixed binomial Poisson cluster process

Assume that the clusters have the form

\begin{equation*} G^{A_i} = \sum_{j=1}^{K_i} \delta_{W_{ij}, A_{ij}}, \end{equation*}

with $$(K_i,(W_{ij})_{j\ge 1},(A_{ij})_{j\ge 0})_{i\ge 0}$$ being an i.i.d. sequence. Assume moreover that the $$(A_{ij})_{j\ge 0}$$ are i.i.d. for any fixed $$i=1,2,\ldots $$ and that $$(A_{ij})_{j\ge1}$$ is independent of $$K_i,(W_{ij})_{j\ge 1}$$ for all $$i\ge 0$$ . We allow for possible dependence between $$K_i,(W_{ij})_{j\ge 1}$$ and the ancestral mark $$A_{i0}$$ ; however, we assume that $$K_i$$ and $$(W_{ij})_{j\ge 1}$$ are conditionally independent given $$A_{i0}$$ . As before, we assume that $${\mathbb{E}}[K]{\lt}\infty$$ . Observe that we use the notation $$W_{ij}$$ instead of $$T_{ij}$$ to emphasize the relatively simple structure of clusters in this model in contrast with the other two models considered in this section. Such a process N is a version of the so-called Neyman–Scott process; see, e.g. Example 6.3(a) of Daley and Vere Jones (Reference Daley and Vere-Jones2003).

2.1.2 Renewal Poisson cluster process

Assume next that the clusters $$G^{A_i}$$ have the distribution

\begin{equation*} G^{A_i} = \sum_{j=1}^{K_i} \delta_{T_{ij}, A_{ij}}, \end{equation*}

where $$(T_{ij})_j$$ represents a renewal sequence

\begin{equation*} T_{ij} = W_{i1} + \cdots + W_{ij}, \end{equation*}

and we keep all other assumptions from the model in Subsection 2.1.1 (in particular, $$(W_{ij})_{j\ge 1}$$ are conditionally i.i.d. and independent of $$K_i$$ given $$A_{i0}$$ ). A general unmarked model of this type is called the Bartlett-Lewis model and analysed in Example 6.3(b) of Daley and Vere Jones (Reference Daley and Vere-Jones2003). See also Faÿ et al. (Reference Faÿ, González-Arévalo, Mikosch and Samorodnitsky2006) for an application of such a point process to the modelling of teletraffic data.

These two simple cluster models were considered by Mikosch (Reference Mikosch2009) in the context of insurance applications. In particular, Subsection 11.3.2 and Example 11.3.5 therein provide expressions for the first two moments of the number of claims in a given time interval [0, t]. Both models can be criticized as overly simple, but the assumption that claims (or delayed payouts) are separated by i.i.d. times (as in the renewal Poisson cluster process) often appears in the risk theory (cf. the Sparre Andersen model; see Asmussen and Albrecher (Reference Asmussen and Albrecher2010)).

2.1.3 Marked Hawkes process

A key motivating example in our analysis is the so-called (linear) marked Hawkes process. Hawkes processes of this type have a neat Poisson cluster representation due to Hawkes and Oakes (Reference Hawkes and Oakes1974). For this model, the clusters $$G^{A}$$ are a recursive aggregation of Cox processes, i.e. Poisson processes with random mean measure $$ \tilde{\mu}_A \times Q$$ , where $$ \tilde{\mu}_A $$ has the form

(2.3) \begin{equation} \tilde{\mu}_{A} (B) = \int_B h(s, A) \,\mathrm{d} s \label{eqn3} \end{equation}

for some fertility (or self-exciting) function h; cf. Example 6.4(c) of Daley and Vere Jones (Reference Daley and Vere-Jones2003).

It is useful to introduce a time shift operator $$\theta_t$$ as

\begin{equation*} \theta _t m = \sum_j \delta_{t_j+t,a_j} \end{equation*}

for an arbitrary point measure $$m = \sum_j \delta_{t_j,a_j} \in M_p$$ and $$t\geq 0$$ . Now, for the ground process $$N^0 \,{=}\, \sum_{i\geq 1} \delta_{{\Gamma_i},A_i}$$ , which is a Poisson point process with intensity $$\nu \times Q$$ on the space $$[0,\infty) \times \mathbb{S}$$ , the cluster process corresponding to a point $$( \Gamma,A)$$ satisfies the recursive relation

(2.4) \begin{equation} G^A = \sum_{l=1}^{{L_A}} ( {\delta_{\tau^1_l,A^1_l} } +\theta_{\tau^1_l} G^{A^1_l} ), \label{eqn4} \end{equation}

where, given A, $$N^A = \sum_{l=1}^{{L_A}} \delta_{\tau^1_l,A^1_l}$$ is a Poisson processes with random mean measure $$ \tilde{\mu}_A \times Q,$$ and the sequence $$(G^{A^1_l})_l$$ is i.i.d., distributed as $$G^A$$ and independent of $$N^A$$ . Thus, at any ancestral point $$(\Gamma,A)$$ a cluster of points appears as a whole cascade of points to the right in time generated recursively according to (2.4). Note that, by definition, $$L_A$$ has Poisson distribution conditionally on A, with mean $$\kappa_A=\int_0^\infty h(s,A) \,\mathrm{d} s$$ . It corresponds to the number of first-generation progeny $$(A^1_l)$$ in the cascade. Note also that the point processes forming the second generation are again Poisson conditionally on the corresponding first generation mark $$A_l^1$$ . The cascade $$G^A$$ corresponds to the process formed by the successive generations, drawn recursively as Poisson processes given the former generation.

The marked Hawkes process is obtained by attaching to the ancestors $$({\Gamma_i},A_{i})$$ of the marked Poisson process $$ N^0 = \sum_{i\geq 1} \delta_{ {\Gamma_i},A_i} $$ a cluster of points, denoted by $$C_i$$ , which contains the point $$(0,A_{i})$$ and a whole cascade $$G^{A_i}$$ of points to the right in time generated recursively according to (2.4) given $$A_i$$ . Under the assumption that

(2.5) \begin{equation} \kappa = {\mathbb{E}} \int h(s,A) \,\mathrm{d} s {\lt} 1, \label{eqn5} \end{equation}

the total number of points in a cluster is generated by a subcritical branching process. Therefore, the clusters are finite a.s., and we denote their size by $$K_i {+1}= C_i[0,\infty)$$ . It is known and not difficult to show that, under (2.5), the clusters always satisfy

\begin{equation*} {\mathbb{E}} K_i {+1} = \frac{1}{1-\kappa}. \end{equation*}

Observe that the clusters $$C_i$$ are independent by construction and can be represented as

$${C_i} = \sum\limits_{j = 0}^{{K_i}} {{\delta _\Gamma }_i + {T_{ij}},Aij} $$

with the $$A_{ij}$$ being i.i.d. and $$T_{i0} = 0$$ . We note that in the case when marks do not influence the conditional density, i.e. when $$h(s,a) = h(s)$$ , the random variable $$K_i{+1}$$ has a so-called Borel distribution with parameter $$\kappa$$ ; see Haight and Breuer (Reference Haight and Breuer1960). Observe also that, in general, marks and arrival times of the final Hawkes process N are not independent of each other, rather, in the terminology of Daley and Vere Jones (Reference Daley and Vere-Jones2003), the marks in the process N are only unpredictable.

Hawkes processes are typically introduced through their conditional intensity. More precisely, a point process $$ N = \sum_{k} \delta_{\tau_k,A^k}$$ represents a Hawkes process of this type if the random marks $$(A^k)$$ are i.i.d. with distribution Q on the space $$\mathbb{S}$$ , while the arrivals $$(\tau_k)$$ have conditional intensity of the form

\begin{equation*} \lambda_t = \lambda(t) = \nu + \sum_{\tau_i{\lt} t} h(t-\tau_i,A^i), \end{equation*}

where $$\nu {\gt} 0$$ is a constant and $$h\colon[0,\infty)\times \mathbb{S}\to \mathbb{R}_+$$ is assumed to be integrable in the sense that $$\int_0^\infty {\mathbb{E}} h(s,A) \,\mathrm{d} s {\lt} \infty$$ . Observe that $$\nu$$ is exactly the constant which determines the intensity of the underlying Poisson process $$N^0$$ due to the Poisson cluster representation of the linear Hawkes processes; cf. Hawkes and Oakes (Reference Hawkes and Oakes1974). Observe that $$\lambda$$ is $$\mathscr{f}_t$$ -predictable, where $$\mathscr{f}_t$$ stands for an internal history of N, $$\mathscr{f}_t = \sigma\{N(I \times S)\colon I \in \mathscr{b}(\mathbb{R}),\, I \subset ({-}\,\infty, t],\, S \in \mathscr{s}\}$$ . Moreover, the $$A^n$$ are assumed to be independent of the past arrival times $$\tau_i,\,i {\lt} n$$ ; see also Bràaud (Reference Brémaud1981). Writing $$N_t = N((0,t] \times \mathbb{S})$$ we can observe that $$(N_t)$$ is an integer-valued process with nondecreasing paths. The role of intensity can be described heuristically by the relation

\begin{equation*} {\mathbb{P}} (\mathrm{d} N_t =1 \ \,\,|\, \ \, \mathscr{f}_{t-}) \approx \lambda_t \,\mathrm{d} t. \end{equation*}

2.1.4 Stationary version

In any of the three examples above, the point process N can be clearly made stationary if we start the construction in (2.1) on the state space $${\mathbb{R}} \times \mathbb{S}$$ with a Poisson process $$ \sum_i \delta_{{\Gamma_i}}$$ on the whole real line. The resulting stationary cluster process is denoted by $$N^*$$ . However, from an applied perspective, it seems more interesting to study the nonstationary version where both the ground process $$N^0$$ and the cluster process itself have arrivals only from some point onwards, e.g. in the interval $$[0,\infty)$$ as, for instance, in Karabash and Zhu (Reference Karabash and Zhu2015).

The stability of various cluster models, i.e. the convergence towards a stationary distribution in an appropriate sense, has been extensively studied for various point processes. For instance, it is known that the unmarked Hawkes process on $$[0,\infty)$$ converges to the stationary version on any compact set and on the positive line under the condition that

(2.6) \begin{equation} \int_0^\infty s h (s) \,\mathrm{d} s {\lt}\infty; \label{eqn6} \end{equation}

see Daley and Vere Jones (2003, p. 232). Using the method of Poisson embedding, originally due to Kerstan (Reference Kerstan1964), Brémaud and Massoulié (Reference Brémaud and Massoulié1996, Section 3) obtained general results on the stability of Hawkes processes, even in the nonlinear case.

4.1 Case $$\alpha \in (1,2)$$

In this case the arguments of the previous section can be adopted to show the following result.

Proposition 4.1

Consider the marked Poisson cluster model introduced in Section 2. Suppose that the $$D_i$$ are regularly varying with index $$\alpha \in (1,2)$$ and that $${\varepsilon}_t = o_P(a_{t}) $$ . Then there exists an $$\alpha$$ -stable random variable $$G_\alpha$$ such that, for $$\mu_D = {\mathbb{E}} D_i$$ ,

(4.2) \begin{equation} \frac{S(t) - t \nu \mu_D }{ a_{\nu t } } \xrightarrow{\text{D}} G_\alpha \quad\text{as } t{\to\infty}. \label{eqn10} \end{equation}

Proof. The proof again follows from (3.1) by an application of Theorem 4.2.6 of Gut (Reference Gut2009) on the random walks $$(\Gamma_n)$$ and $$(S^D_n)$$ together with (4.1). By assumption, we have $${\varepsilon}_t/a_{\nu t}\sim \nu^{-1/\alpha} {\varepsilon}_t /a_t {\xrightarrow{{\mathbb{P}\ }}} 0$$ . To complete the proof, we observe that the sequences $$({\Gamma_n})$$ and $$(D_n)$$ are independent; hence,

\begin{equation*} \dfrac{D_{\tau(t)}}{a_{\nu t}} \xrightarrow{{\mathbb{P}\ }} 0 \quad\text{as } t\to \infty. \end{equation*}

4.2 Case $$\alpha \in (0,1)$$

In this case we were not able to find any result of Anscombe’s theorem type for two-dimensional random walks of the type used above. Therefore, as our initial step, we prove a theorem which we believe is new and of independent interest. It concerns partial sums of i.i.d. nonnegative regularly varying random variables, say $$(Y_n)$$ , subordinated to an independent renewal process. More precisely, set $$V_n= Y_1+\cdots + Y_n,\, n\geq 1$$ . Suppose that the sequence $$(Y_n)$$ is independent of another i.i.d. sequence of nonnegative and nontrivial random variables $$(W_n)$$ . Denote by

$$\sup \,\emptyset = 0$$

the corresponding renewal process, where we set $$\sup \varnothing = 0$$ . Recall that, for regularly varying random variables $$Y_i$$ , there exists a sequence $$(a_n)$$ such that $$n {\mathbb{P}} (Y_i{\gt}a_n) \to 1$$ as $$n {\to\infty}$$ . The limiting behaviour of the process $$V_{\sigma(t)}$$ was considered by Anderson (Reference Anderson1988) in the case where the $$W_i$$ are themselves regularly varying with index $$\leq 1.$$

Since $${\sigma(t)}/{t} {\xrightarrow{\text{a.s.}}} \nu,$$ if $$0{\lt}{\mathbb{E}} W_i = {1}/{\nu} {\lt} \infty,$$ we may expect that $$V_{\sigma(t)}$$ has similar asymptotic behaviour as $$V_{\nu t}$$ for $$t \rightarrow \infty$$ . It is not too difficult to make this argument rigorous if, for instance, $${\mathbb{E}} W_i^2 {\lt} \infty,$$ because then $$ ( \sigma(t) - t \nu )^2/t,\, t{\gt}0$$ , is uniformly integrable by Gut (2009, Section 2.5). The following functional limit theorem gives a precise description of the asymptotic behaviour of $$V_{\sigma(t)}$$ whenever $$W_i$$ has a finite mean.

Theorem 4.1

Suppose that $$(Y_n)$$ and $$(W_n)$$ are independent nonnegative i.i.d. sequences of random variables such that the $$Y_i$$ are regularly varying with index $$\alpha \in (0,1)$$ , and such that $$ 0{\lt} 1/\nu = {\mathbb{E}} W_i {\lt} \infty $$ . Then in the space $$D[0,\infty)$$ endowed with the Skorokhod’s $$J_1$$ topology

(4.3) \begin{equation} \frac{V_{\sigma(t\cdot)} }{a_{ \nu t } } \xrightarrow{\text{D}} G_\alpha (\cdot)\quad\text{as } t\to \infty, \label{eqn11} \end{equation}

where $$(G_\alpha(s))_{s\geq 0}$$ is an $$\alpha$$ -stable subordinator.

Proof. Since the $$Y_i$$ are regularly varying, it is known (Resnick (Reference Resnick1987), (Reference Resnick2007)) that the point process convergence

(4.4) \begin{equation} M'_n= \sum_i \delta_{i/n, {Y_i}/{a_n}} \xrightarrow{\text{D}} M_{\alpha} \sim {\rm PRM} ({\rm Leb} \times d ({-}\, y^{-\alpha})) \label{eqn12} \end{equation}

holds as $$n {\to\infty}$$ with respect to the vague topology on the space of Radon point measures on $$[0,\infty)\times (0,\infty]$$ . The abbreviation PRM stands for the Poisson random measure, indicating that the limit is a Poisson process. Starting from (4.4), it was shown in Resnick (Reference Resnick2007, Chapter 7) for instance that, for an $$\alpha$$ -stable subordinator $$G_\alpha(\cdot)$$ as in the statement of the theorem,

\begin{equation*} V'_n (\cdot) = \frac{V_{\lfloor n \cdot \rfloor}}{a_n} \xrightarrow{\text{D}} G_\alpha (\cdot)\quad\text{as } n\to \infty, \end{equation*}

in the Skorokhod’s $$J_1$$ topology on the space $$ D[0,\infty)$$ . Observe that, since $$\alpha \in (0,1)$$ , no centering is needed, and we can substitute the integer index n by a continuous index $$t {\to\infty}$$ . Note further that we have the joint convergence

(4.5) \begin{equation} (M'_t,V'_t) \xrightarrow{\text{D}} (M_\alpha, G_\alpha)\quad\text{as } t\to \infty, \label{eqn13} \end{equation}

in the product topology on the space of point measures and càdlàg functions. Moreover, it is known that the jump times and sizes of the $$\alpha$$ -stable subordinator $$G_\alpha$$ correspond to the points of the limiting point process $$M_{\alpha}$$ .

The space of point measures and the space of càdlàg functions $$D[0,\infty)$$ are both Polish, in respective topologies; therefore, Skorokhod’s representation theorem applies. Thus, we can assume that the convergence in (4.5) holds a.s. on a certain probability space $$(\Omega,\mathscr{f},P)$$ , and in particular there exists $$\Omega' \subseteq \Omega$$ such that $$P(\Omega') = 1$$ and, for all $$\omega \in \Omega'$$ , $$V'_t \to G_\alpha $$ in $$J_1$$ and $$M'_t \to M_\alpha$$ in vague topology. By Theorem 2.15 of Jacod and Shiryaev (2003, Chapter VI), for any such $$\omega$$ , there exists a dense set $$B=B(\omega)$$ of points in $$[0,\infty)$$ such that

\[ {V'_t} (s) \to G_\alpha (s)\quad\text{as } t\to \infty \]

for every $$s \in B $$ , where actually B is simply the set of all nonjump times in the path of the process $$G_\alpha $$ . On the other hand, it is known that in the $$J_1$$ topology, on some set $$\Omega''$$ such that $$P(\Omega'') = 1$$ ,

(4.6) \begin{equation} \frac{\sigma(t\cdot)}{t\nu} \to {\rm id}(\cdot)\quad\text{as } t\to \infty, \label{eqn14} \end{equation}

where id stands for the identity map. This follows directly by an application of Theorem 2.15 of Jacod and Shiryaev (2003, Chapter VI). Moreover, by Proposition VI.1.17 of Jacod and Shiryaev (Reference Jacod and Shiryaev2003), the convergence in (4.6) holds locally uniformly on $$D[0,\infty)$$ .

Now consider, for fixed $$t{\gt}0$$ and $$\omega \in \Omega' \cap \Omega''$$ ,

\[ V_t (s) = \dfrac{V_{\sigma(t s) }}{a_{\nu t}}, \qquad s \geq 0. \]

From (4.6) we may expect that $$V_t (s) \approx V'_{t\nu } (s).$$ Indeed, for any fixed $$0{\lt}\delta{\lt}1$$ and all large t, we know that $$ \lfloor tc\nu (1-\delta) \rfloor\leq\sigma(tc) \leq \lfloor tc\nu (1+\delta) \rfloor$$ , which by monotonicity of the sums implies that

\[ \dfrac{V_{\lfloor tc\nu (1-\delta) \rfloor} }{a_{\nu t}} \leq\dfrac{V_{\sigma(t c)} }{a_{\nu t}} \leq\dfrac{V_{\lfloor tc\nu (1+\delta) \rfloor } }{a_{\nu t}}. \]

Now, for $$c(1-\delta) $$ and $$c(1+\delta)$$ in B, the left-hand side and the right-hand side above converge to $$G_\alpha(c(1-\delta))$$ and $$G_\alpha(c(1+\delta))$$ . Thus, if we consider $$c\in B$$ and let $$\delta \to 0$$ then

(4.7) \begin{equation} \dfrac{V_{\sigma(t c)} }{a_{\nu t}} \to G_\alpha(c)\quad\text{as } t\to \infty \label{eqn15} \end{equation}

for all $$\omega \in \Omega' \cap \Omega''$$ and thus with probability 1.

By Theorem 2.15 of Jacod and Shiryaev (2003, Chapter VI), to prove (4.3), it remains to show that, for all $$\omega \in \Omega' \cap \Omega''$$ and $$c\in B$$ , as $$t {\to\infty}$$ ,

(4.8) \begin{equation} \sum_{0{\lt}s\leq c} | \Delta V_t(s)|^2 = \sum_{i{\lt}\sigma(tc) } \bigg(\frac{Y_i}{a_{t\nu}}\bigg)^2 \to \sum_{0{\lt}s\leq c} | \Delta G_\alpha(s)|^2 , \label{eqn16} \end{equation}

where, for an arbitrary càdlàg process X(t) at time $$t \geq 0$$ , we define $$\Delta X(t) = X_{t} - X_{t-} $$ . Observe that

\begin{equation*} G_{\alpha/2}(c)\,{:\!=}\sum_{0{\lt}s\leq c} | \Delta G_\alpha(s)|^2 \end{equation*}

defines an $$\alpha/2$$ -stable subordinator and that the squared random variables $$Y_i^2$$ are again regularly varying with index $$\alpha/2$$ with the property that $$n\ {\mathbb{P}}(Y_i^2 {\gt}a^2_n) \to 1$$ . A similar approximation argument as for (4.7) shows that (4.8) indeed holds, which concludes the proof.

Assume now that $$P(D{\gt}x)= x^{-\alpha} \ell(x)$$ for some slowly varying function $$\ell$$ and $$\alpha \in(0,1)$$ . Select a sequence $$a_n {\to\infty}$$ such that $$n\, P(D{\gt}a_n) \to 1$$ as $$n{\to\infty}$$ . Under suitable conditions on the residue term $${\varepsilon}_t$$ , we obtain the following result.

Proposition 4.2

Assume that the $$D_i$$ are regularly varying with index $$\alpha \in (0,1)$$ and that $${\varepsilon}_t \,{=}\, o_P(a_{t}) $$ . Then there exists an $$\alpha$$ -stable random variable $$G_\alpha$$ such that

(4.9) \begin{equation} \frac{S(t) }{ a_{\nu t } } \xrightarrow{\text{D}} G_\alpha\quad\text{as }t{\to\infty}. \label{eqn17} \end{equation}

Proof. The proof follows roughly the same lines as the proof of Proposition 4.1, but here we rely on an application of the previous theorem to the random walks $$(\Gamma_n)$$ and $$(S^D_n)$$ . However, instead of the $$Y_i$$ and $$W_i$$ we have the $$D_i$$ and an independent sequence of i.i.d. exponential random variables with parameter $$\nu$$ .

Remark 4.1

One can consider the total claim amount in the period [0, t] for the stationary model of Subsection 2.1.4, i.e.

\[ S^*(t) = \int_{0}^t \int_\mathbb{S} f(u) N^* (\mathrm{d} s,\,\mathrm{d} u),\qquad t \geq 0. \]

Here, again, $$S^*(t)$$ has a similar representation as in (3.1) but with an additional term on the right-hand side, i.e.

\begin{equation*} S^*(t) = \sum_{i=1}^{\tau(t)} D_i - D_{\tau(t)} - {{\varepsilon}}_t + {{\varepsilon}}^*_{0,t},\qquad t \geq 0, \end{equation*}

where

\begin{equation*} {\varepsilon}^*_{0,t} = \sum_{\Gamma_i \leq 0 ,\, 0 {\lt} \Gamma_i+T_{ij}{\lt}t} X_{ij}. \end{equation*}

Clearly, by stationarity,

(4.10) \begin{equation} {\varepsilon}_t = \sum_{0\leq {\Gamma_i} \leq t ,\, t {\lt} {\Gamma_i}+T_{ij}} X_{ij} \stackrel{\text{D}}{=} {\varepsilon}^{-}_t = \sum_{-t\le {\Gamma_i} \leq 0 ,\, 0 {\lt} {\Gamma_i}+T_{ij}} X_{ij}. \label{eqn18} \end{equation}

Hence, $${\varepsilon}_t = o_P(a_{t}) $$ yields $${\varepsilon}^-_t = o_P(a_{t}) $$ for any sequence $$(a_t)$$ and, therefore,

\[ {\varepsilon}^*_{0,t} \leq {\varepsilon}_t ^-+ \sum_{\Gamma_i {\lt} -t ,\, 0 {\lt} \Gamma_i+T_{ij}{\lt}t} X_{ij}= \sum_{\Gamma_i {\lt} -t ,\, 0 {\lt} \Gamma_i+T_{ij}{\lt}t} X_{ij}+o_P(a_{t}). \]

In particular, the conclusions of Propositions 3.1, 4.1, and 4.2 hold for random variables $$S^*(t)$$ also under the additional assumption that

(4.11) \begin{equation} \tilde {\varepsilon}_t\,{:\!=}\sum_{\Gamma_i \leq -t ,\, 0 {\lt} \Gamma_i+T_{ij}{\lt}t} X_{ij} =o_P(a_{t}). \label{eqn19} \end{equation}

5.1 Mixed binomial cluster model

Recall from Subsection 2.1.1 that the clusters in this model have the form

\begin{equation*} G^{A_i} = \sum_{j=1}^{K_i} \delta_{W_{ij}, A_{ij}}. \end{equation*}

Assume that

• • $${({K_i},\,{({W_{ij}})_{j \ge 1}},\,{({A_{ij}})_{j \ge 0}})_{i \ge 0}}$$ constitutes an i.i.d. sequence,

• • $$(A_{ij})_{j\ge0}$$ are i.i.d. for any fixed i,

• • $$(A_{ij})_{j\ge1}$$ is independent of $$K_i,(W_{ij})_{j\ge 1}$$ for all $$i\ge 0$$ ,

• • $$(W_{ij})_{j\ge 1}$$ are conditionally i.i.d. and independent of $$K_i$$ given $$A_{i0}$$ .

Thus, we do not exclude the possibility of dependence between $$K_i,(W_{ij})_{j\ge 1}$$ and the ancestral mark $$A_{i0}$$ . For any $$\gamma{\,\gt\,}0$$ , we denote by

\[ A,\ X_j,\ K, W_j,\ m_A, \ m_A^{(\gamma)}, \]

generic random variables with the same distribution as $$A_{ij}$$ , $$X_{ij}=\,f(A_{ij})$$ , $$K_i$$ , $$W_{ij}$$ , $${\mathbb{E}}[K_i\ \,\,|\, \ \, A_{i0}]$$ , and $${\mathbb{E}}[K_i^\gamma\ \,\,|\, \ \, A_{i0}]$$ , respectively. Using the cluster representation, we can derive the asymptotic properties of S(t). Let us first consider the Gaussian central limit theorem under appropriate second moment assumptions. Denote by $${\mathbb{P}}(W \in \cdot \ \,\,|\, \ \, A)$$ the distribution of the $$W_{ij}$$ given $$A_{i0}$$ .

Corollary 5.1. Assume that $${\mathbb{E}}[X^2]{\lt}\infty$$ and $${\mathbb{E}}[K^2]{\lt}\infty$$ . If

(5.1) \begin{equation} \sqrt t {\mathbb{E}}[m_A{\mathbb{P}}(W{>}t\ \,\,|\, \ \, A)] \to 0\quad\text{as } t\to \infty, \label{eqn20} \end{equation}

then relation (3.2) holds.

Observe that (5.1) is slightly weaker than the existence of the moment $${\mathbb{E}}[K \sqrt W]{\lt}\infty$$ .

Proof of Corollary 5.1. It follows from the compound sum representation of D that $${\mathbb{E}} D^2\,{{\lt}}\,\infty$$ as soon as $${\mathbb{E}}[X^2]{\lt}\infty$$ and $${\mathbb{E}}[K^2]{\lt}\infty$$ . By Proposition 3.1, it remains to show that $${\varepsilon}_t \,{=}\, o_P(\sqrt t)$$ . To this end, we use the Markov inequality

\[ {\mathbb{P}}({\varepsilon}_t{>}\sqrt t)\le \frac{{\mathbb{E}}[{\varepsilon}_t]}{\sqrt t}=\frac{{\mathbb{E}}[\!\sum_{0\le \Gamma_i\le t}\sum_{j=1}^{K_i}{{\bf 1}}_{\{t{\lt} \Gamma_i+W_{ij}\}}\,f(A_{ij})]}{\sqrt t}. \]

We use Lemma 7.2.12 of Mikosch (Reference Mikosch2009) with $$f(s) = \sum_{j=1}^{K_i}{{\bf 1}}_{\{W_{ij}{>} t-s\}} \,f(A_{ij})$$ in order to compute the right-hand side term as

\begin{align*} \frac{\int_0^t{\mathbb{E}}[\!\sum_{j=1}^{K_i}{{\bf 1}}_{\{W_{ij}{>} t-s\}} \,f(A_{ij})]\nu \,\mathrm{d} s}{\sqrt t} &=\frac{\nu{\mathbb{E}}[X]\int_0^t{\mathbb{E}}[{\mathbb{E}}[\! \sum_{j=1}^{K_i}{{\bf 1}}_{\{W_{ij}{>} t-s\}} \ \,\,|\, \ \, A_{i0}]]\,\mathrm{d} s}{\sqrt t} \\ &=\frac{\nu{\mathbb{E}}[X]\int_0^t{\mathbb{E}}[m_A{\mathbb{P}}(W{>} x\ \,\,|\, \ \, A)]\,\mathrm{d} x}{\sqrt t}. \end{align*}

Note that the last identity follows due to the independence of $$K_i$$ and $$(W_{ij})_{j\ge 0}$$ conditionally on $$A_{i0}$$ . We conclude by l’Hôpital’s rule that this converges to 0 under (5.1).

For regularly varying D of order $$1{\lt}\alpha{\lt}2$$ , we obtain the corresponding limit theorem under weaker assumptions on the tail of the waiting time W.

Corollary 5.2. Assume that one of the (RV) conditions holds for $$1{\lt}\alpha{\lt}2$$ , so that D is regularly varying. When

(5.2) \begin{equation} t^{1+\delta-1/\alpha} {\mathbb{E}}[m_A{\mathbb{P}}(W{>}t\ \,\,|\, \ \, A)] \to 0\quad \text{as }t\to \infty \label{eqn21} \end{equation}

for some $$\delta{>}0$$ , relation (4.2) holds.

Condition 5.2 is slightly weaker than assuming that $${\mathbb{E}}[m_A W^{1+\delta-1/\alpha}]{\lt}\infty$$ . Note that, when $$\alpha\to 1^+$$ and K is independent of W, this condition boils down to the existence of a $$\delta$$ th moment of W for any strictly positive $$\delta$$ .

Proof of Corollary 5.2. By definition, $$(a_t)$$ satisfies $$t{\mathbb{P}}(D{>}a_t)\to 1$$ as $$t\to \infty$$ and $$(a_t)$$ is regularly varying with index $$1/\alpha$$ . Applying the Markov inequality as in the proof of Corollary 5.1, we obtain

\[ {\mathbb{P}}({\varepsilon}_t{>}a_t)\le \frac{{\mathbb{E}}[{\varepsilon}_t]}{a_t}=\frac{\nu{\mathbb{E}} [X]\int_0^t{\mathbb{E}}[m_A{\mathbb{P}}(W{>} s\ \,\,|\, \ \, A)]\,\mathrm{d} s}{a_t}. \]

The claim follows by l’Hôpital’s rule and the relation $$t^{1/\alpha-\delta} = o(a_t)$$ for any $$\delta{>}0$$ .

Remark 5.1

In the context of the mixed binomial model, consider the total claim amount of the stationary process denoted by $$S^*(t)$$ which also takes into account the arrivals in the interval $$({-}\,\infty,0)$$ ; see Remark 4.1. Assume for simplicity that the $$K_i$$ and $$(W_{ij})$$ are unconditionally independent. Then $$\tilde {\varepsilon}_t$$ from (4.11) is $$o_P(a_{t})$$ under the same conditions as in Corollaries 5.1 and 5.2, where we set $$a_t = \sqrt t$$ in the former case. Indeed, we will show that

\begin{equation*} {\mathbb{E}} \tilde {\varepsilon}_t = {\mathbb{E}} \bigg[\sum_{\Gamma_i \leq -t ,\, 0 {\lt} \Gamma_i+W_{ij}{\lt}t} X_{ij}\bigg] = {\mathbb{E}} \bigg[\sum_{-t\le {\Gamma_i} \leq 0 ,\, t {\lt} {\Gamma_i}+W_{ij}} X_{ij} \bigg], \end{equation*}

so that $${\mathbb{E}} \tilde {\varepsilon}_t= o_P(a_{t})$$ as well since the right-hand side is dominated by $${\mathbb{E}} {\varepsilon}^-_t = o(a_t)$$ ; cf. 4.10.

Note first that, under the assumptions of the last two corollaries, individual claims have finite expectation, i.e. $${\mathbb{E}} X {\lt}\infty$$ . So it suffices to show that

\begin{equation*} I_1 \,{:\!=}\, \frac{{\mathbb{E}} \tilde {\varepsilon}_t}{ {\mathbb{E}} X} = {\mathbb{E}} \sum_{\Gamma_i {\lt} -t } \sum_{j=1}^{K_i} {{\bf 1}}_{\{ 0 {\lt} \Gamma_i+W_{ij}{\lt}t\}} = {\mathbb{E}} \sum_{-t{\lt}\Gamma_i {\lt} 0 } \sum_{j=1}^{K_i} {{\bf 1}}_{\{\Gamma_i+W_{ij}{>}t\}} \,{=\!:}\, I_2 . \end{equation*}

From $$I_1$$ and $$I_2$$ we respectively subtract the left-hand and right-hand sides of the equality

\[ {\mathbb{E}} \sum_{-2t{\lt}\Gamma_i {\lt} -t } \sum_{j=1}^{K_i} {{\bf 1}}_{\{\Gamma_i+W_{ij} \in (0,t) \}} {{\bf 1}}_{\{W_{ij}\in (t,2t]\}} = {\mathbb{E}} \sum_{-t{\lt}\Gamma_i {\lt} 0 } \sum_{j=1}^{K_i} {{\bf 1}}_{\{\Gamma_i+W_{ij} \in (t,2t)\} } {{\bf 1}}_{\{W_{ij}\in (t,2t]\}}, \]

where the equality follows by the stationarity of the underlying Poisson process, to obtain

\begin{equation*} J_1 = {\mathbb{E}} \sum_{-\infty{\lt}\Gamma_i {\lt} -t } \sum_{j=1}^{K_i} {{\bf 1}}_{\{0{\lt}\Gamma_i+W_{ij} {\lt} t\} } {{\bf 1}}_{\{W_{ij} {>} 2t \}} = {\mathbb{E}} K \int_{-\infty}^{-t} \nu \,\mathrm{d} s \int_{-s \vee 2t }^{t-s} \,\mathrm{d} F_W(u) \end{equation*}

and

\begin{equation*} J_2 = {\mathbb{E}} \sum_{-t {\lt}\Gamma_i {\lt} 0 } \sum_{j=1}^{K_i} {{\bf 1}}_{\{\Gamma_i+W_{ij} {>} t \}} {{\bf 1}}_{\{W_{ij} {>} 2t \}} = {\mathbb{E}} K \int_{-t}^{0} \nu \,\mathrm{d} s \int_{2t}^\infty \,\mathrm{d} F_W(u), \end{equation*}

where $$F_W$$ denotes the distribution function of delays $$(W_{ij})$$ . Finally, note that

\begin{equation*} J_1 = {\mathbb{E}} K \int_{ 2t }^{\infty} \,\mathrm{d} F_W(u) \int_{-u}^{t-u} \nu \,\mathrm{d} s = J_2. \end{equation*}

Since we assumed that $${\mathbb{E}}[K]{\lt}\infty$$ , the regular variation property of D with index $$\alpha\in(0,1)$$ can arise only through the claim size distribution; see Proposition 4.8 of Faÿ et al. (Reference Faÿ, González-Arévalo, Mikosch and Samorodnitsky2006). It turns out that in such a heavy-tailed case, no additional assumption on the waiting time W is needed.

Corollary 5.3. Assume that X is regularly varying of order $$0{\lt}\alpha{\lt}1$$ . Then relation 4.9 holds.

Proof. Observe that we cannot apply the Markov inequality anymore because $${\mathbb{E}} D=\infty$$ . Instead, we use the fact that $$\sum\nolimits_{j = 1}^t {{X_j}/{a_t}} $$ converges because X and D have equivalent regularly varying tails. Recall from (4.10) that

\begin{equation*} {\varepsilon}_t = \sum_{0\le {\Gamma_i} \leq t ,\, t {\lt} {\Gamma_i}+W_{ij}} X_{ij}. \end{equation*}

We denote the (increasing) number of summands in the right-hand side term by $$M_t=\# \{i,j\colon 0\le {\Gamma_i} \leq t ,\, t {\lt} {\Gamma_i} + W_{ij}\}$$ . We can apply Proposition 4.2 after observing that $$\sum\nolimits_{j=1}^{M_t}X_j/a_{M_t}$$ is a tight family of random variables, because $$M_t$$ is independent of the array $$(X_{ij})$$ . Writing

(5.3) \begin{equation} \frac{{\varepsilon}_t}{a_t}\stackrel{\text{D}}{=}\frac{\sum_{j=1}^{M_t}X_j}{a_{M_t}}\frac{a_{M_t}}{a_t}, \label{eqn22} \end{equation}

and observing that $$a_t$$ is regularly varying with index $$1/\alpha$$ , we obtain the desired result provided that $$M_t=o_P(t)$$ . It is sufficient to show the convergence to 0 of the ratio

\begin{align*} \frac{{\mathbb{E}} [M_t]}{t}=\frac{{\mathbb{E}} [\# \{i,j\colon 0\le {\Gamma_i} \leq t ,\, t{\lt} {\Gamma_i} + W_{ij}\}]}{t} =\frac{{\mathbb{E}}[\!\sum_{0\le {\Gamma_i} \leq t }\sum_{j=1}^{K_i}{{\bf 1}}_{\{t\le {\Gamma_i} +W_{ij}\}}]}{t}. \end{align*}

Using similar calculations as in the proof of Corollary 5.1 (setting $$X=1$$ ), we obtain an explicit formula for the right-hand side term as

\[ \frac{\nu \int_0^t{\mathbb{E}}[m_A{\mathbb{P}}(W{>}x\ \,\,|\, \ \, A)]\,\mathrm{d} x}{t}\to 0 \quad\text{as } t\to \infty, \]

the convergence to 0 following from a Cesarò argument.

5.2 Renewal cluster model

Recall from Subsection 2.1.2 that the clusters of this model have the form

\begin{equation*} G^{A_i} = \sum_{j=1}^{K_i} \delta_{T_{ij}, A_{ij}}, \end{equation*}

where

• • $$T_{ij}=W_{i1}+\cdots +W_{ij}$$ ,

• • while $$(K_i,(W_{ij})_{j\ge 1},(A_{ij})_{j\ge 0})_{i\ge 0}$$ constitutes an i.i.d. sequence satisfying the assumptions listed in Section 5.1.

The total claim amount coming from the ith immigrant and its progeny is again

\[ D \stackrel{\text{D}}{=} \sum_{j=0}^K X_j \]

for $$(X_j)$$ i.i.d. copies of f(A). Dealing with the waiting times $$T_{ij}=W_{i1}+\cdots +W_{ij}$$ requires additional care than in the previous model. We first obtain the following result.

Corollary 5.4. Suppose that $${\mathbb{E}} X^2 {\lt}\infty,$$ $${\mathbb{E}} K^2 {\lt}\infty$$ , and $$ {\mathbb{E}} [K^{2}W^\delta] {\lt} \infty$$ for some $$\delta{>}\tfrac12$$ . Then relation (3.2) holds.

Proof. The proof follows from Proposition 3.1. The second moment of the $$D_i$$ is finite by the moment assumptions on X and K. It remains to show that the residue term satisfies $${\varepsilon}_t = o_P(\sqrt{t}) $$ . Using Lemma 7.2.12 of Mikosch (Reference Mikosch2009) with $$f(x) = \sum\nolimits_{j=1}^{K_i}{{\bf 1}}_{\{W_{i1}+\cdots+W_{ij}{\gt} x\}} \,f(A_{ij})$$ similarly as in the proof of Corollary 5.1, we obtain

\begin{align*} {\mathbb{E}}[{\varepsilon}_t]&= \int_0^t{\mathbb{E}}\bigg[\sum_{j=1}^{K_i}{{\bf 1}}_{\{W_{i1}+\cdots+W_{ij}{>} x\}} \,f(A_{ij})\bigg]\nu \,\mathrm{d} x \\ & = \nu \int_0^t{\mathbb{E}}\bigg[{\mathbb{E}}\bigg[\sum_{j=1}^{K_i} {{\bf 1}}_{\{W_{i1}+\cdots+W_{ij}{>} x\}} \,f(A_{ij})\biggm| K_i, (W_{ij})_{j\ge 1}\bigg]\bigg] \,\mathrm{d} x \\ & = \nu {\mathbb{E}}[X]\int_0^t{\mathbb{E}}\bigg[ \sum_{j=1}^{K_i}{{\bf 1}}_{\{W_{i1}+\cdots+W_{ij}{>} x\}} \bigg] \,\mathrm{d} x \end{align*}

by the independence between $$K_i, (W_{ij})_{j\ge 1}$$ and $$(\,f(A_{ij}))_{j\ge 1}$$ . The key argument in dealing with the renewal cluster model is the upper bound

(5.4) \begin{equation} \sum_{j=1}^{K_i}{{\bf 1}}_{\{W_{i1}+\cdots+W_{ij}{>}x\}}\le {{\bf 1}}_{\{ W_{i1}+\cdots+W_{iK_i}{>}x\}} K_i. \label{eqn23} \end{equation}

Assume without loss of generality that $$\delta \le 1$$ . By the Markov inequality and the conditional independence of $$K_i$$ and $$(W_{ij})_{j\ge 0}$$ conditionally on $$A_{i0}$$ , we obtain

(5.5) \begin{align} {\mathbb{E}}[ {{\bf 1}}_{W_{i1}+\cdots+W_{iK_i}{>} x}K_i\ \,\,|\, \ \, A_{i0} ]&\le \frac{{\mathbb{E}}[K_i (W_{i1}+\cdots+W_{iK_i})^\delta\ \,\,|\, \ \, A_{i0}]}{x^\delta}\nonumber \\ &\le m_{A_{i0}}^{(2)} \frac{{\mathbb{E}}[W^\delta\ \,\,|\, \ \, A_{i0}]}{x^\delta}, \label{eqn24}\end{align}

using the notation $$m_{A_{i0}}^{(\gamma)}={\mathbb{E}}[K_i^\gamma \ \,\,|\, \ \, A_{i0}]$$ for any $$\gamma{>}0$$ . The last inequality follows from the sublinearity of the mapping $$x\mapsto x^\delta$$ for $$\delta\le 1$$ . Thus, we obtain, for some constant $$C{>}0$$ ,

\[ {\mathbb{E}}[{\varepsilon}_t]\le \nu {\mathbb{E}}[X] {\mathbb{E}}[m_A^{(2)} W^\delta]\int_1^tx^{-\delta}\,\mathrm{d} x +C=O({\mathbb{E}}[m_A^{(2)} W^\delta]t^{1-\delta})= o(\sqrt t) \]

as $$\delta{>}\tfrac12$$ by assumption.

Regularly varying claims can be handled with additional care as K may not be square integrable.

Corollary 5.5. Assume that one of the (RV) conditions holds so that D is regularly varying of order $$1{\lt}\alpha{\lt}2.$$ Suppose further that $${\mathbb{E}}[K^{1+\gamma}]{\lt}\infty$$ and $${\mathbb{E}}[K^{1+\gamma}W^\delta]{\lt}\infty,$$ $$\delta{>}0,$$ $$\gamma {>}0$$ , and $$ \delta{\gt}(\alpha-\gamma)/\alpha$$ . Then relation (4.2) holds.

Observe that we obtain somewhat stronger conditions than in the mixed binomial case; see Corollary 5.2 and the remark following it.

Proof of Corollary 5.5. Without loss of generality we assume that $$\gamma\le1$$ . We use the Markov inequality of order $$\gamma$$

\[ {\mathbb{P}}({\varepsilon}_t{>}a_t)\le \frac{{\mathbb{E}}[{\varepsilon}_t^\gamma]}{a_t^\gamma}. \]

Thanks to the subadditivity of the function $$x\mapsto x^\gamma$$ we have

\begin{align*} {\mathbb{E}}[{\varepsilon}_t^\gamma]&= {\mathbb{E}}\bigg[\bigg(\sum_{0\le \Gamma_i\le t}\sum_{j=1}^{K_i}{{\bf 1}}_{\{\Gamma_i+W_{i1} +\cdots+W_{ij}{>}t\}}\,f(A_{ij})\bigg)^\gamma\bigg] \\ &\le {\mathbb{E}}\bigg[\sum_{0\le \Gamma_i\le t}\bigg(\sum_{j=1}^{K_i}{{\bf 1}}_{\{\Gamma_i+ W_{i1}+\cdots+W_{ij}{>}t\}}\,f(A_{ij})\bigg)^\gamma\bigg]. \end{align*}

Using Lemma 7.2.12 of Mikosch (Reference Mikosch2009) with $$f(x) = \sum\nolimits_{j=1}^{K_i}{{\bf 1}}_{\{W_{i1}+\cdots+W_{ij}{\gt}x\}}\,f(A_{ij})$$ similarly as in the proof of Corollary 5.1, we obtain

(5.6) \begin{equation} {\mathbb{E}}[{\varepsilon}_t^\gamma]\le\nu \int_0^t{\mathbb{E}}\bigg[ \bigg(\sum_{j=1}^{K_i}{{\bf 1}}_{\{W_{i1}+\cdots+W_{ij}{>}x\}}\,f(A_{ij})\bigg)^\gamma\bigg] \,\mathrm{d} x. \label{eqn25} \end{equation}

Using Jensen’s inequality,

\begin{align*} &{\mathbb{E}}\bigg[{\mathbb{E}}\bigg[ \bigg(\sum_{j=1}^{K_i}{{\bf 1}}_{\{W_{i1}+\cdots+W_{ij}{>}x\}} \,f(A_{ij})\bigg)^\gamma\biggm| K_i, (W_{ij})_{j\ge 1}\bigg]\bigg] \\ &\qquad\le {\mathbb{E}}\bigg[ \bigg({\mathbb{E}}\bigg[\sum_{j=1}^{K_i}{{\bf 1}}_{\{W_{i1}+\cdots+W_{ij}{>}x\}}\,f(A_{ij})\biggm| K_i, (W_{ij})_{j\ge 1}\bigg]\bigg)^\gamma\bigg], \end{align*}

so that, using the independence between $$K_i, (W_{ij})_{j\ge 1}$$ and $$(\kern1.9pt f(A_{ij}))_{j\ge 1}$$ , we obtain

\[ {\mathbb{E}}[{\varepsilon}_t^\gamma]\le \nu {\mathbb{E}}[X]^\gamma\int_0^t{\mathbb{E}}\bigg[\bigg(\sum_{j=1}^{K_i}{{\bf 1}}_{\{W_{i1}+\cdots+W_{ij}{>}x\}}\bigg)^\gamma\bigg]\,\mathrm{d} x. \]

Using the stochastic domination (5.4), we obtain

\[ {\mathbb{E}}[{\varepsilon}_t^\gamma]\le\nu{\mathbb{E}}[X]^\gamma \int_0^t{\mathbb{E}}[ {{\bf 1}}_{ \{W_{i1}+\cdots+W_{iK_i}{>}x\}}K_i^\gamma]\,\mathrm{d} x. \]

Without loss of generality we assume that $$0{\lt}\delta{\lt}1$$ . Applying the Markov inequality of order $$\delta$$ conditionally on $$A_{i0}$$ as in (5.5), we have

\[ {\mathbb{E}}[{{\bf 1}}_{ \{W_{i1}+\cdots+W_{iK_i}{>}x\}}K_i^\gamma\ \,\,|\, \ \, A_{i0}]\le m_{A_{i0}}^{(1+\gamma)} \frac{{\mathbb{E}}[W_i^\delta\ \,\,|\, \ \, A_{i0}]}{x^\delta}. \]

Substituting the previous inequality into this bound we obtain, for some $$C{>}0$$ ,

\[ {\mathbb{E}}[{\varepsilon}_t^\gamma]\le\nu{\mathbb{E}} [X]^\gamma{\mathbb{E}}[m_{A}^{(1+\gamma)}W^\delta]t^{1-\delta}+C =o(a_t^\gamma) \]

as $$1-\delta{\lt}\gamma/\alpha$$ by assumption.

Corollary 5.6. If X is regularly varying of order $$\alpha \in (0,1)$$ then relation 4.9 holds.

Proof. We use the same arguments as in the proof of Corollary 5.1 in order to obtain 5.3. The desired result follows if we can show that $$M_t=\# \{i,j\colon 0\le {\Gamma_i} \leq t ,\, t {\lt} {\Gamma_i} + W_{i1} + \cdots + W_{ij}\} = o_P(t)$$ . Using Markov’s inequality, it is enough to check that $${\mathbb{E}}[M_t]/t=o(1)$$ . Following the same reasoning as in the proof of Corollary 5.5, we estimate the moment of $$M_t$$ similarly as that of $${\varepsilon}_t$$ in (5.6):

\[ {\mathbb{E}}[M_t]\le \int_0^t{\mathbb{E}}\bigg[ \sum_{j=1}^{K_i}{{\bf 1}}_{\{W_{i1}+\cdots+W_{ij}{>}x\}} \bigg]\nu \,\mathrm{d} x\le \int_0^t{\mathbb{E}}[ K_i {{\bf 1}}_{\{W_{i1}+\cdots+W_{iK_i}{>}x\}} ]\nu \,\mathrm{d} x. \]

We again used stochastic domination (5.4) to obtain the last upper bound. From a Cesaró argument, the result will follow if

\[ {\mathbb{E}}[ K_i {{\bf 1}}_{\{W_{i1}+\cdots+W_{iK_i}{>}x\}} ] \to 0\quad\text{as } x\to \infty. \]

We can check this negligibility property because the random sequence $$K_i {{\bf 1}}_{\{W_{i1}+\cdots+W_{iK_i}{>}x\}} \to 0$$ a.s. by the finiteness of $$W_{i1}+\cdots+W_{iK_i}$$ and because the sequence is dominated by $$K_i$$ that is integrable.

5.3 Marked Hawkes process

Recall from Section 2.1.3 that the clusters of the Hawkes model satisfy recursive relation (2.4). In other words, the clusters $$G^{A_i}$$ represent a recursive aggregation of Poisson processes with random mean measure $$ \tilde{\mu}_A \times Q$$ which satisfies $$\kappa = {\mathbb{E}} \int h(s,A) \,\mathrm{d} s {\lt} 1$$ .

In general, it is not entirely straightforward to see when the moments of D are finite. However, note that the $$D_i$$ are i.i.d. and satisfy the distributional equation

(5.7) \begin{equation} D \stackrel{\text{D}}{=} f(A) + \sum_{j=1}^{L_A} D_j , \label{eqn26} \end{equation}

where $$L_A$$ has the Poisson distribution conditionally on A, with mean $$\kappa_A=\int_0^\infty h(s,A) \,\mathrm{d} s.$$ Recall from (2.5) that $$\kappa = {\mathbb{E}} \kappa_A {\lt} 1.$$ The $$D_j$$ on the right-hand side are independent of $$\kappa_A$$ and i.i.d. with the same distribution as D. Conditionally on A, the waiting times are i.i.d. with common density $$h(t,A)/\kappa_A ,\,t\ge0$$ . Thus, we can relate the clusters of the Hawkes process with those of a mixed binomial process from Section 5.1 with $$K=L_A$$ . In order to obtain the asymptotic properties of S(t), we still need to characterize the moment and tail properties of D.

Consider the Laplace transform of D, i.e. $$\varphi(s) = {\mathbb{E}} \mathrm{e}^{-sD}$$ for $$s \geq 0$$ . Also, recall that the Laplace transform of a Poisson compound sum is of the form

\begin{equation*} {\mathbb{E}} [ \mathrm{e}^{-s \sum_{j=1}^{M}Z_j} ] = {\mathbb{E}} [ \mathrm{e}^{m_A ( {\mathbb{E}} \mathrm{e}^{-sZ} - 1)} ], \end{equation*}

where M is Poiss$$(m_A)$$ distributed, independent of the i.i.d. sequence $$(Z_i)$$ of nonnegative random variables with common distribution (see, for instance, Section 7.2.2 of Mikosch (Reference Mikosch2009)). Note that $$\varphi$$ is an infinitely differentiable function for $$s{\gt}0$$ . To simplify the notation, let

\[ X= f(A), \]

a generic claim size and observe that, by (5.7), $$\varphi$$ satisfies

(5.8) \begin{align} \varphi(s) & = {\mathbb{E}} [ {\mathbb{E}}[ \mathrm{e}^{-s(X + \sum_{j=1}^{L_A}D_j)} \ \,\,|\, \ \, A ]]\nonumber \\ &= {\mathbb{E}} [ \mathrm{e}^{-sX} {\mathbb{E}}[\mathrm{e}^{ -s \sum_{j=1}^{L_A}D_j} \ \,\,|\, \ \, A ]] \nonumber \\ & = {\mathbb{E}} [ \mathrm{e}^{-sX} \mathrm{e}^{\kappa_A ({\mathbb{E}} \mathrm{e}^{-sD} -1) } ] \nonumber \\ &= {\mathbb{E}}[ \mathrm{e}^{-sX} \mathrm{e}^{\kappa_A (\varphi(s) -1) }]. \label{eqn27}\end{align}

When $${\mathbb{E}} [\kappa_A ]=\kappa{\lt}1$$ , it is known that this functional equation has a unique solution $$\varphi$$ which further uniquely determines the distribution of D. By studying the behaviour of the derivatives of $$\varphi(s)$$ for $$s\to 0+$$ , we obtain the following result.

Lemma 5.1

If $${\mathbb{E}} X^2 {\lt}\infty$$ and $${\mathbb{E}} \kappa_A^2{\lt}\infty$$ then

\[ {\mathbb{E}} D^2=\frac{{\mathbb{E}} {X^2}}{1-\kappa}+\frac{({\mathbb{E}}{X})^2}{(1-\kappa)^3}{\mathbb{E}} \kappa_A ^2+2\frac{{\mathbb{E}} X}{(1-\kappa)^2}{\mathbb{E}} (X\kappa_A ){\lt}\infty. \]

Note that this expression coincides with the expression in [Karabash and Zhu (Reference Karabash and Zhu2015), when $$X=f(A)\equiv 1$$ , i.e. in the case when one simply counts the number of claims.

Proof of Lemma 5.1. Differentiating (5.8) with respect to $$s{\gt}0$$ produces

\[ \varphi'(s) = {\mathbb{E}} [ \mathrm{e}^{-sX} \mathrm{e}^{\kappa_A (\varphi(s) -1) } ({-}\,X + \kappa_A \varphi' (s) )]. \]

As $${\mathbb{E}}(\kappa_A)=\kappa{\lt}1$$ , we obtain

(5.9) \begin{equation} \varphi'(s) = \frac{-{\mathbb{E}} [ \mathrm{e}^{-sX} \mathrm{e}^{\mu (\varphi(s) -1) } X ]}{1-{\mathbb{E}}[ \kappa_A ^{-sX} \mathrm{e}^{\kappa_A (\varphi(s) -1) }\kappa_A ]}. \label{eqn28} \end{equation}

As $$\varphi(s)\le 1,\,s\ge 0$$ , the integrand in the numerator is dominated by X and that in the denominator by $$\kappa_A $$ . By the dominated convergence argument, $$\lim _{s \to 0+}\varphi'(s)$$ exists and is equal to

\[ \varphi'(0) = \dfrac{- {\mathbb{E}} X}{1-\kappa}. \]

In particular, $${\mathbb{E}} D = {\mathbb{E}} X/(1-\kappa)$$ . Differentiating (5.8) again produces the second moment of D. Indeed, we have

\[ \varphi''(s) = {\mathbb{E}} [ \mathrm{e}^{-sX} \mathrm{e}^{\kappa_A (\varphi(s) -1) } (({-}\,X + \kappa_A \varphi' (s) )^2 + \kappa_A \varphi'' (s) )] , \]

so that

(5.10) \begin{equation} \varphi''(s) = \frac{{\mathbb{E}} [ \mathrm{e}^{-sX} \mathrm{e}^{\kappa_A (\varphi(s) -1) } ({-}\,X + \kappa_A \varphi' (s) )^2]}{1-{\mathbb{E}}[\mathrm{e}^{-sX} \mathrm{e}^{\kappa_A (\varphi(s) -1) }\kappa_A ]}. \label{eqn29} \end{equation}

Here, again, applying the dominated convergence theorem twice, we can let $$s\to 0+$$ and obtain

\[ \varphi''(0) = \dfrac{{\mathbb{E}} ( {-}\,X + \kappa_A \varphi' (0) )^2}{1-\kappa}=\dfrac{{\mathbb{E}} ( X + \kappa_A {\mathbb{E}} D )^2}{1-\kappa}. \]

This completes the proof since $$X\,{=}\ f(A)$$ .

The following theorem describes the behavior of the total claim amount (S(t)) for the marked Hawkes process under appropriate second moment assumptions. Recall from (2.3) that $$\tilde{\mu}_{A} (B) = \int_B h(s, A) \,\mathrm{d} s.$$

Theorem 5.1. If $$\kappa{\lt}1$$ , $${\mathbb{E}} X^2 {\lt}\infty$$ , and $${\mathbb{E}}[ \kappa_A ^2]{\lt}\infty$$ then, in either the stationary or nonstationary case, if

(5.11) \begin{align} \sqrt t {\mathbb{E}}[\tilde{\mu}_A(t,\infty)]\to 0\quad\text{as } t\to \infty, \label{eqn30} \end{align}

relation (3.2) holds.

Proof. In order to apply Proposition 3.1, we have to check that $${\varepsilon}_t = o_P(\sqrt t)$$ . The proof is based on the following domination argument on $${\varepsilon}_t$$ . Recall that we can write

\begin{equation*} N = \sum_i \sum_j \delta_{{\Gamma_i} + T_{ij},A_{ij}} = \sum_{k=1}^\infty \delta_{\tau_k,A^{k}} , \end{equation*}

without loss of generality assuming that $$0\leq \tau_1\leq \tau_2\leq \cdots$$ . At each time $$\tau_j$$ , a claim arrives generated by one of the previous claims or an entirely new (immigrant) claim appears. In the former case, if $$\tau_j$$ is a direct offspring of a claim at time $$\tau_i$$ , we write $$\tau_i\to \tau_j$$ . Progeny $$\tau_j$$ then creates potentially further claims. We denote by $$D_{\tau_j}$$ the total amount of claims generated by the arrival at $$\tau_j$$ (counting the claim at $$\tau_j$$ itself as well). Clearly, the $$D_{\tau_j}$$ are identically distributed as D and even independent if we consider claims which are not offspring of one another. They are also independent of everything happening in the past.

The process N is naturally dominated by the stationary marked Hawkes process $$N^*$$ which is well defined on the whole real line as we assumed that $$\kappa = {\mathbb{E}} \kappa_A {\lt}1$$ ; see the discussion at the end of Subsection 2.1. For the original and stationary Hawkes processes, N and $$N^*$$ , by $$\lambda$$ and $$ \lambda^*$$ we denote the corresponding predictable intensities. By the construction of these two point processes, $$\lambda \leq \lambda^*$$ . Recall that $$\tau_i\to \tau_j$$ is equivalent to $$\tau_j=\tau_i+W_{ik},\,k\le L^i = L_{A^i}$$ , where, by assumption, the $$W_{ik}$$ are i.i.d. with common density $$h(t,A^i)/ \kappa_{A^i},\,t\ge0$$ , and independent of $$L^{i}$$ conditionally on the mark $$A^i$$ of the claim at $$\tau_i$$ . Moreover, conditionally on $$A^i$$ , the number of direct progeny of the claim at $$\tau_i$$ , denoted by $$L^{i}$$ , has Poisson distribution with parameter $$\tilde{\mu}_{A^i}$$ . Therefore, using conditional independence and equal distribution of the $$D^{'}$$ we obtain

\begin{align*} {\mathbb{E}}[{\varepsilon}_t] &= {\mathbb{E}}\bigg[ \sum_{{\Gamma_i} \leq t}\sum_j {{\bf 1}}_{\{\Gamma_i + T_{ij} {\gt}t\}} X_{ij}\bigg] \\ &= {\mathbb{E}}\bigg[\sum_{\tau_i \leq t} \sum_{\tau_j {\gt} t} D_{\tau_j} {{\bf 1}}_{\{\tau_i\to \tau_j\}}\bigg] \\ &={\mathbb{E}}\bigg[\sum_{\tau_i \leq t} {\mathbb{E}}\bigg[\sum_{k=1}^{L^{i}} D_{\tau_i+W_{ik}} {{\bf 1}}_{\{\tau_i+W_{ik} {\gt} t\}} \biggm| (\tau_i,A^i)_{i\ge 0}; \,\tau_i \le t\bigg]\bigg] \\ &= \mu_D {\mathbb{E}} \bigg[ \int_{0}^t \int_{\mathbb{S}} \tilde{\mu}_a((t-s,\infty)) N(\mathrm{d} s,\,\mathrm{d} a) \bigg], \end{align*}

where $$ \tilde{\mu}_{a} ((u,\infty)) = \int_u^\infty h(s, a) \,\mathrm{d} s$$ . Observe that, from the projection theorem (see Brémaud (Reference Brémaud1981, Chapter 8, Theorem 3)), the last expression equals

\[ \mu_D {\mathbb{E}}\bigg[\int_0^t \int_{\mathbb{S}} \tilde{\mu}_a((t-s,\infty)) Q(\mathrm{d} a) \lambda(s)\,\mathrm{d} s \bigg]. \]

We can further bound this estimate by

\begin{align*} {\mathbb{E}}\bigg[\int_0^t \int_{\mathbb{S}} \tilde{\mu}_a((t-s,\infty)) Q(\mathrm{d} a) \lambda^*(s)\,\mathrm{d} s\bigg]&=\int_0^t \int_{\mathbb{S}} \tilde{\mu}_a((t-s,\infty)) Q(\mathrm{d} a) {\mathbb{E}}[\lambda^*(s)]\,\mathrm{d} s \\ &=\frac{\nu}{1-\kappa}\int_0^t \int_{\mathbb{S}} \tilde{\mu}_a((t-s,\infty)) Q(\mathrm{d} a) \,\mathrm{d} s. \end{align*}

Here we used Fubini’s theorem, and the expression $${\mathbb{E}} [ \lambda^*(s) ] \equiv \nu /(1-\kappa)$$ . Observe that this expectation is constant since $$N^*$$ is a stationary point process; to show that it equals $$ \nu/(1-\kappa)$$ , note that

\begin{align*}\\[-24pt] {\mu}^* &= {\mathbb{E}} { \lambda^*(s)} \\ &= {\mathbb{E}} \bigg[ \nu + \int_{-\infty}^s \int_{\mathbb{S}} h(s-u,a) N^*(\mathrm{d} u,\,\mathrm{d} a) \bigg] \\ & = \nu + \int_{-\infty}^s \int_{\mathbb{S}} h(s-u,a) {\mathbb{E}}({\lambda}^*(u))\,\mathrm{d} uQ(\mathrm{d} a)[0] \\ & = \nu + {\mu}^* \int_{-\infty}^s {\mathbb{E}} h(s-u,A)\,\mathrm{d} u \\ & = \nu + {\mu}^*\int_0^{\infty}{\mathbb{E}} h(v,A)\,\mathrm{d} v; \end{align*}

see also Daley and Vere Jones (Reference Daley and Vere-Jones2003, Example 6.3(c)). Hence, $${\mu}^* = \nu + {\mu}^* \kappa$$ and $$ {\mu}^* = {\nu}/({1-\kappa})$$ , as claimed above. Now, we have

(5.12) \begin{equation} {\mathbb{E}} {{\varepsilon}_t} \leq \dfrac{\nu }{1-\kappa} \int_0^t \int_{\mathbb{S}} \tilde{\mu}_a((t-s,\infty)) Q(\mathrm{d} a)\,\mathrm{d} s = \dfrac{\nu }{1-\kappa} \int_0^t \mu_D \int_s^\infty {\mathbb{E}}[h(u,A)]\,\mathrm{d} u \,\mathrm{d} s . \label{eqn31} \end{equation}

Hence, the residual term is bounded in expectation by the expression we obtained in the mixed binomial case in Section 5.1. Thus, the result will follow from the proof of Corollary 5.1 under condition (5.1), which is further equivalent to (5.11) thanks to the expression of the density of the waiting times.

Dividing the last expression by $$\sqrt t$$ and applying l’Hôpital’s rule proves the theorem for the nonstationary or pure Hawkes process; see [Karabash and Zhu (Reference Karabash and Zhu2015) where the same idea appears in the proof of Theorem 1.3.2.

To show that the central limit theorem holds in the stationary case, note that S(t) now has a similar representation as in (3.1) but with an additional term on the right-hand side, i.e.

(5.13) \begin{equation} S(t) = \sum_{i=1}^{\tau(t)} D_i - D_{\tau(t)} - {\varepsilon}_t +{\varepsilon}_{0,t},\qquad t \geq 0, \label{eqn32} \end{equation}

where

\begin{equation*} {\varepsilon}_{0,t} = \sum_{\Gamma_i \leq 0 ,\, 0 {\lt} \Gamma_i+T_{ij}{\lt}t} X_{ij} \, . \end{equation*}

A similar computation yields

\begin{align*} {\mathbb{E}} {{\varepsilon}_{0,t}}&= {\mathbb{E}} \sum_{\Gamma_i \leq 0} \sum_j {{\bf 1}}_{\{0{\lt}\Gamma_i + T_{ij} {\lt}t\}} X_{ij} \\ &= {\mathbb{E}} \sum_{\tau_i \leq 0} \sum_{0{\lt} \tau_j{\lt}t} D_{\tau_j} {{\bf 1}}_{\{\tau_i\to \tau_j\}} \\ &= {\mathbb{E}} \bigg[ \sum_{\tau_i \leq 0} \mu_D {\mathbb{E}} \bigg[ \sum_{0{\lt}\tau_j{\lt}t} {{\bf 1}}_{\{\tau_i\to \tau_j\}} \biggm| \mathscr{f}_0 \bigg] \bigg] \\ & = \mu_D {\mathbb{E}} \bigg[ \sum_{\tau_i \leq 0} \tilde{\mu}_{A^{i}}((0-\tau_i,t-\tau_i)) \bigg] \\ & = \mu_D {\mathbb{E}} \bigg[ \int_{-\infty}^0 \int_{\mathbb{S}} \tilde{\mu}_a(({-}\,s,t-s)) N^*(\mathrm{d} s,\,\mathrm{d} a) \bigg], \end{align*}

where $$\tilde{\mu}_{a} (B) = \int_B h(s, a) \,\mathrm{d} s$$ and $$\mathscr{f}_0$$ stands for the internal history of the process up to time 0, i.e. $$\mathscr{f}_0 = \sigma\{ N(I \times S)\colon I \in \mathscr{b}(\mathbb{R}),\, I \subset ({-}\,\infty, 0],\, S \in \mathscr{s}\}.$$ Again, by the projection theorem (see Brémaud (Reference Brémaud1981, Chapter 8, Theorem 3)), the last expression equals

\begin{eqnarray*} \mu_D {\mathbb{E}} \bigg[ \int_{-\infty}^0 \int_{\mathbb{S}} \tilde{\mu}_a(({-}\,s,t-s)) \lambda^*(s)\,\mathrm{d} s\,Q(\mathrm{d} a) \bigg]. \end{eqnarray*}

This is further equal to

\begin{align*} &\mu_D \int_{-\infty}^0 \int_{\mathbb{S}} \tilde{\mu}_a(({-}\,s,t-s)) {\mathbb{E}} [ \lambda^*(s) ] \,\mathrm{d} s\,Q(\mathrm{d} a) \\ &\qquad = \mu_D \dfrac{\nu}{1-\kappa} \int_{-\infty}^0 \int_{\mathbb{S}} \tilde{\mu}_a(({-}\,s,t-s)) \,\mathrm{d} s\,Q(\mathrm{d} a) \\ &\qquad = \mu_D \dfrac{\nu}{1-\kappa} \int_{-\infty}^0 {\mathbb{E}} \tilde{\mu}_A(({-}\,s,t-s)) \,\mathrm{d} s \\ &\qquad = \mu_D \dfrac{\nu}{1-\kappa} \int_{0}^{\infty} {\mathbb{E}} \tilde{\mu}_A((s,s+t)) \,\mathrm{d} s \\ &\qquad = \mu_D \dfrac{\nu}{1-\kappa} \int_0^{\infty} {\mathbb{E}} \int_s^{s+t} h(u,A)\,\mathrm{d} u \,\mathrm{d} s \\ &\qquad = \mu_D\dfrac{\nu}{1-\kappa} \int_0^{\infty} {\mathbb{E}} (t\wedge u) h(u,A) \,\mathrm{d} u \\ &\qquad= \mu_D\dfrac{\nu}{1-\kappa} \bigg( \int_0^{t} {\mathbb{E}} [ u h(u,A) ] du+t\int_t^{\infty} {\mathbb{E}} [ h(u,A) ] \,\mathrm{d} u \bigg). \end{align*}

Note that the second term in the last expression divided by $$\sqrt t$$ tends to 0 by (5.11). Using integration by parts for the first term, we have

\begin{equation*} \int_0^{t} {\mathbb{E}} [ u h(u,A) ] \,\mathrm{d} u = t \int_t^\infty {\mathbb{E}}[h(s,A)]\,\mathrm{d} s+\int_0^t\int_u^\infty {\mathbb{E}}[h(s,A)]\,\mathrm{d} s\,\mathrm{d} u. \end{equation*}

The first integral on the right-hand side divided by $$\sqrt t$$ tends to 0 under (5.11). The last term divided by $$\sqrt t$$ also tends to 0 by an application of l’Hôpital’s rule as in the nonstationary case.

Finally, we observe that $${\varepsilon}_{0,t}/\sqrt t \xrightarrow{{\mathbb{P}\ }} 0$$ and the result in the stationary case is proved.

Observe that (5.11) is substantially weaker than (2.6) in the unmarked case. Namely, the former condition requires only that the total residue due to the claims on the compact interval [0,t] is of the order $$o(\sqrt t)$$ in probability. In particular, in the unmarked case, the central limit theorem holds for the stationary and the nonstationary cases even if (2.6) is not satisfied, i.e. even when the nonstationary process is not convergent.

As mentioned above, there are related limit theorems in the literature concerning only the counting process $$N_t$$ (see Karabash and Zhu (Reference Karabash and Zhu2015)), but in contrast to their result, our proof does not rely on the martingale central limit theorem, it stems from the rather simple relations (3.1) and (5.13).

In the following example, we consider some special cases of Hawkes processes for which a closed-form expression for the second moment $${\mathbb{E}} D^2$$ can be found.

Example 5.1. (Marked Hawkes processes with claims independent of the cluster size.)

Assume that the random measure (2.3),

\begin{equation*} \tilde{\mu}_A (B) = \int_B h(s, A) \,\mathrm{d} s, \end{equation*}

holds on $$\mathbb{R}_+$$ and that the corresponding claim sizes $$X=f(A)$$ are independent. In particular, this holds if $$\tilde{\mu}_A (B) = \int_B h(s) \,\mathrm{d} s$$ for some integrable function h, i.e. when $$\tilde{\mu}_A$$ is a deterministic measure and we actually have a standard Hawkes process with independent marks. In this special case $$K+1$$ is known to have the so-called Borel distribution; see Haight and Breuer (Reference Haight and Breuer1960).

Using the arguments from the proof of Lemma 5.1, we obtain $$\mu_D = {\mathbb{E}} D_i = {{\mathbb{E}} X}/({1-\kappa}).$$ Similarly, the variance of the $$D_i$$ is finite as the variance of a compound sum, and equals

\begin{equation*} \sigma_D^2 = \dfrac{{\sigma_X^2}}{1-\kappa} + \dfrac{\kappa {{({\mathbb{E}} X)}^2} }{(1-\kappa)^3}; \end{equation*}

cf. Lemma 2.3.4 of Mikosch (Reference Mikosch2009). Hence, $${\mathbb{E}} D^2 $$ in Theorem 5.1 has the form

\begin{equation*} {\mathbb{E}} D^2 = \sigma_D^2 + \mu_D^2 = {\frac{\sigma_X^2}{1-\kappa} + \frac{{({\mathbb{E}} X)}^2}{(1-\kappa)^3}}. \end{equation*}

In the special case when the claims are all constant, say $$X=f(A)\equiv c{\gt}0$$ , direct calculation yields $${\mathbb{E}} D = c/({1-\kappa}),$$ with $$\kappa={\mathbb{E}} [\kappa_A ]$$ , and

\[ \varphi''(0) = \dfrac{{\mathbb{E}} ({-}\,c+ \kappa_A \varphi' (0) )^2}{1-\kappa} = {c^2}\dfrac{\mathop{\rm var}\nolimits \kappa_A + 1}{(1-\kappa)^3}, \]

obtaining

\[ {\mathbb{E}} D^2 = \varphi''(0) ={c^2} \dfrac{\mathop{\rm var}\nolimits \kappa_A + 1}{(1-\kappa)^3}, \]

in particular, for $$c=1$$ , we recover expression in Karabash and Zhu (Reference Karabash and Zhu2015).

In the rest of this subsection, we study the marked Hawkes process in the case when the $$D_i$$ are regularly varying with index $$\alpha {\lt}2 $$ . Using the result of Hult and Samorodnitsky (Reference Hult and Samorodnitsky2008), we can show that, when the individual claims $$X=f(A)$$ are regularly varying, this property is frequently passed on to the random variable D under appropriate moment assumptions on $$\kappa_A $$ . However, using the specific form of the Laplace transform for D given in (5.8), we can show regular variation of D under weaker conditions. This is the content of the following lemma.

Theorem 5.2. The nonnegative random variable X is regularly varying with a noninteger tail index $$\alpha{\gt}0$$ , i.e. $$\skew4\bar F(x)\sim x^{-\alpha}\ell(x)$$ as $$x\to\infty$$ if and only if

\[ \varphi^{(\lceil \alpha \rceil)}(s) \sim cs^{\alpha-\lceil \alpha \rceil}\ell\bigg(\frac{1}{s}\bigg)\quad\text{as } s\to 0+ \]

for some slowly varying function $$\ell$$ and a constant depending only on $$\alpha$$ : $$c=-\Gamma(\alpha+1)\Gamma(1-\alpha)/\Gamma(\alpha-\lfloor \alpha \rfloor)$$ .

Consider first the case $$0{\lt}\alpha{\lt}1$$ . By differentiating once the expression for the Laplace transform, we obtain identity (5.9),

\[ \varphi'(s) =\frac{-{\mathbb{E}} [ \mathrm{e}^{-sX} \mathrm{e}^{\kappa_A (\varphi(s) -1) } X ]}{1-{\mathbb{E}}[\mathrm{e}^{-sX} \mathrm{e}^{\kappa_A (\varphi(s) -1) }\kappa_A ] },\qquad s{\gt}0. \]

We are interested in the behavior of this derivative as $$s\to 0+$$ . Using the inequality $$|1-\mathrm{e}^{-x}|=1-\mathrm{e}^{-x}\le x$$ , we have

\begin{align*} \bigg|\varphi'(s)-\frac{-{\mathbb{E}} [ \mathrm{e}^{-sX} X ]}{1-{\mathbb{E}}[\mathrm{e}^{-sX-\kappa_A (1-\varphi(s))}\kappa_A ] }\bigg|&\le \frac{{\mathbb{E}} [ \kappa_A (1-\varphi(s)) \mathrm{e}^{-sX} X ]}{1-{\mathbb{E}}[\mathrm{e}^{-sX-\kappa_A (1-\varphi(s))}\kappa_A ] } \\ &\le \frac{1-\varphi(s) }s\frac{{\mathbb{E}} [\kappa_A \mathrm{e}^{-sX} sX ]}{1-{\mathbb{E}}[\mathrm{e}^{-sX-\kappa_A (1-\varphi(s))}\kappa_A ] }. \end{align*}

As $$\mathrm{e}^{-sX} sX\le \mathrm{e}^{-1},$$ we prove that $${\mathbb{E}} [\kappa_A \mathrm{e}^{-sX} sX ]=o(1)$$ as $$s\to 0+$$ by dominated convergence. As in the proof of Lemma 5.1, the denominator $$1-{\mathbb{E}}[\mathrm{e}^{-sX-\kappa_A (1-\varphi(s))}\kappa_A ]$$ is controlled thanks to dominated convergence as well. Moreover, again using $$1-\mathrm{e}^{-x}\le x$$ and denoting by $$\varphi_X(s)={\mathbb{E}}[\mathrm{e}^{-sX}]$$ the Laplace transform of X, we have

\[ 0\le \varphi_X(s)-\varphi(s)\le {\mathbb{E}}[\mathrm{e}^{-sX}\kappa_A (1-\varphi(s)) ]\le \kappa (1-\varphi(s)), \]

so that

\[ 1-\varphi(s)\le \frac1{1-\kappa}(1-\varphi_X(s)). \]

Collecting all these bounds and using the identity $$\varphi'_X(s)=-{\mathbb{E}}[\mathrm{e}^{-sX}X]$$ , we obtain

(5.14) \begin{equation} \bigg|\varphi'(s)-\frac{\varphi'_X(s)}{1-{\mathbb{E}}[\text{e}^{-sX} \text{e}^{\kappa_A (\varphi(s) -1) }\kappa_A ]} \bigg| =o\bigg(\frac{1-\varphi_X(s)}s\bigg)\quad\text{as } s\to 0^+. \label{eqn33} \end{equation}

The regular variation of the random variable D now follows from the regular variation of the random variable X by two consecutive applications of Theorem 5.2. First, as X is regularly varying of order $$0{\lt}\alpha{\lt}1$$ , applying the direct part of the equivalence in Theorem 5.2 yields

\[ \varphi_X'(s)\sim c s^{\alpha -1 }\ell\bigg(\frac{1}{s}\bigg)\quad\text{as } s\to 0^+. \]

Applying Karamata’s theorem, i.e. the equivalence between Equations (8.1.9) and (8.1.11b) of Bingham et al. (Reference Bingham, Goldie and Teugels1987, Theorem 8.1.6), we obtain $$(1-\varphi_X(s))/s=O(\varphi'_X(s))$$ as $$s\to 0+$$ . Using (5.14) and the limiting relation

\begin{equation*} {\mathbb{E}}[\mathrm{e}^{-sX} \mathrm{e}^{\kappa_A (\varphi(s) -1) }\kappa_A ]\to \kappa\quad\text{as } s\to 0^+, \end{equation*}

we obtain

\[ \varphi'(s)\sim \frac{\varphi'_X(s)}{1-\kappa}\sim \frac{c s^{\alpha -1 }\ell(1/s)}{1-\kappa}\quad\text{as } s\to 0^+. \]

Finally, applying the reverse part of Theorem 5.2, we obtain

\[ \skew4\bar F_D( x)\sim \frac{\ell(x)x^{-\alpha }}{1-\kappa}=\frac{\skew4\bar F_X(x)}{1-\kappa}\quad\text{as } x\to \infty. \]

The case $$1{\lt}\alpha{\lt}2$$ can be treated similarly, under the additional assumption that $$Y=X + \kappa_A \mu_D$$ is regularly varying. We will again show that $${P}(D{\gt}x)\sim (1-\kappa)^{-1}{P}(Y{\gt}x)$$ as $$x\to \infty$$ . To prove this equivalence, recall identity (5.10),

\[ \varphi''(s) = \frac{{\mathbb{E}} [ \mathrm{e}^{-sX} \mathrm{e}^{\kappa_A (\varphi(s) -1) } (\!-\!X + \kappa_A \varphi' (s) )^2]}{1-{\mathbb{E}}[\mathrm{e}^{-sX} \mathrm{e}^{\kappa_A (\varphi(s) -1) }\kappa_A ]}. \]

As $$\alpha{\gt}1$$ , we have that $${\mathbb{E}}[Y]{\lt}\infty$$ and, thus, $$ {\mathbb{E}}[X]{\lt}\infty$$ and $$\mathbb{E}[D]=\mu_D=(1-\kappa)^{-1}{\mathbb{E}}[X]$$ . Observe that, for any $$s{\gt}0$$ ,

\begin{align*} &\bigg|\varphi''(s)-\frac{{\mathbb{E}} [ \mathrm{e}^{-sY} Y^2 ]}{1-{\mathbb{E}}[\mathrm{e}^{-sX-\kappa_A (1-\varphi(s))}\kappa_A ] }\bigg| \\ &\qquad=\bigg|\frac{{\mathbb{E}}[\mathrm{e}^{-sX-\kappa_A (1-\varphi(s) ) } ( {-}\,X + \kappa_A \varphi' (s) )^2]-{\mathbb{E}} [ \mathrm{e}^{-sY} Y^2 ]}{1-{\mathbb{E}}[\mathrm{e}^{-sX-\kappa_A (1-\varphi(s))}\kappa_A ] }\bigg| . \end{align*}

Let us decompose the numerator into two terms:

\begin{align*} \underbrace{{\mathbb{E}}|\mathrm{e}^{-sX-\kappa_A (1-\varphi(s))} ( ( {-}\,X + \kappa_A \varphi' (s) )^2-Y^2)|}_{I_1} +\underbrace{{\mathbb{E}}|( \mathrm{e}^{-sY}-\mathrm{e}^{-sX-\kappa_A (1-\varphi(s))} ) Y^2 |}_{I_2} . \end{align*}

Using the identity $$a^2-b^2=(a-b)(a+b)$$ , $$I_1$$ is bounded by

\begin{align*} I_1&\le (\mu_D + \varphi'(s)){\mathbb{E}}[\mathrm{e}^{-sX-\kappa_A (1-\varphi(s))}\kappa_A (2X+\kappa_A (\mu_D-\varphi'(s)))] \\ &\le \frac{\mu_D + \varphi'(s)}s({\mathbb{E}}[2\kappa_A \mathrm{e}^{-sX}sX]+{\mathbb{E}}[\kappa_A \mathrm{e}^{-\kappa_A (1-\varphi(s))}s(\mu_D-\varphi'(s))]). \end{align*}

As $$\mathrm{e}^{-sX}sX\le \mathrm{e}^{-1}$$ , then $${\mathbb{E}}[2\kappa_A \mathrm{e}^{-sX}sX]=o(1)$$ as $$s\to 0^+$$ by dominated convergence. By the convexity of $$\varphi(s)$$ we have $$1-\varphi(s)\ge -\varphi'(s)s$$ for any $$s{\gt}0$$ . Thus,

\[ \mathrm{e}^{-\kappa_A (1-\varphi(s))}({-}\,\varphi'(s)s)\le \mathrm{e}^{-\kappa_A ({-}\,\varphi'(s)s)}({-}\,\varphi'(s)s)\le \mathrm{e}^{-1} \]

and the dominated convergence argument also applies to the second integrand as $$-\varphi'(s)s\le 1-\varphi(s)=o(1)$$ . We obtain $$I_1=o((\mu_D + \varphi'(s))/s)$$ as $$s\to 0^+$$ . In order to control the rate of $$\left( {{\mu _D} + \varphi '(s)} \right)/s$$ , we note that $$\varphi(s)$$ is $$\mu_D$$ Lipschitz on $$s\ge0$$ so that $$|1-\varphi(s)|=1-\varphi(s)\le \mu_D s$$ . Then

\[ sX+\kappa_A (1-\varphi(s))\le sX+s\kappa_A \mu_D =sY, \]

and we bound

\begin{align*} \varphi'(s)\le \frac{-{\mathbb{E}}[\mathrm{e}^{-sY}X]}{1-{\mathbb{E}}[\mathrm{e}^{-sY }\kappa_A ]} \le \frac{\varphi_Y'(s)}{1-{\mathbb{E}}[\mathrm{e}^{-sY }\kappa_A ]}+\frac{\kappa\mu_D}{1-{\mathbb{E}}[\mathrm{e}^{-sY }\kappa_A ]}, \end{align*}

where $$\varphi_Y(s)={\mathbb{E}}[\mathrm{e}^{-sY}]$$ denotes the Laplace transform of Y. It yields the estimates $$\mu_D + \varphi'(s)=O(\mu_D + \varphi_Y'(s))+ O(\kappa-{\mathbb{E}}[\mathrm{e}^{-sY }\kappa_A ])$$ . Again using the fact that $$1-\mathrm{e}^{-x}\le x$$ on the second term, we obtain $$I_1=o((\mu_D + \varphi_Y'(s))/s)+o(1)$$ as $$s\to 0^+$$ .

We now turn to the term $$I_2$$ that we identify as

\begin{align*} I_2&= {\mathbb{E}} | (\mathrm{e}^{sX+\kappa_A (1-\varphi(s))-sY}-1) \mathrm{e}^{-sX-\kappa_A (1-\varphi(s))} Y^2 | \\ & = {\mathbb{E}} | (\mathrm{e}^{-\kappa_A (\mu_D s- (1-\varphi(s)))}-1) \mathrm{e}^{-sX-\kappa_A (1-\varphi(s))} Y^2 | . \end{align*}

As $$1-\varphi(s)\le \mu_D s$$ , the term in the absolute value is negative for $$s{\gt}0$$ and

\[ I_2= {\mathbb{E}} [ (1-\mathrm{e}^{-\kappa_A (\mu_D s- (1-\varphi(s)))}) \mathrm{e}^{-sX-\kappa_A (1-\varphi(s))} Y^2 ] . \]

Again using the basic inequality $$1-\mathrm{e}^{-x}\le x$$ for $$x\ge 0 $$ , we obtain the new estimate

\begin{align*} I_2&\le {\mathbb{E}} [ \kappa_A (s\mu_D-(1-\varphi(s)) \mathrm{e}^{-sX-\kappa_A (1-\varphi(s))}Y^2 ] \\ &\le \frac{s\mu_D-(1-\varphi(s))}{s^2}{\mathbb{E}} [ \kappa_A \mathrm{e}^{-sX-\kappa_A (1-\varphi(s))}(sY)^2 ]. \end{align*}

We have

\[ (sY)^2\le (sX +\kappa_A (1-\varphi(s)))^2 + \kappa_A^2(s\mu_D-(1-\varphi(s)))^2. \]

As $$\mathrm{e}^{-x} x^2\le 4\mathrm{e}^{-2}$$ for any $$x{\gt}0$$ , we prove that

\[ {\mathbb{E}} [\kappa_A \mathrm{e}^{-(sX+\kappa_A (1-\varphi(s))}(sX+\kappa_A (1-\varphi(s)))^2]=o(1) \]

as $$s\to 0^+$$ by dominated convergence. It remains to bound the term

\[ \mathrm{e}^{-sX-\kappa_A (1-\varphi(s))} \kappa_A^2(s\mu_D-(1\,{-}\,\varphi(s)))^2 \]

uniformly for $$s{\gt}0$$ sufficiently small. As $$1-\varphi(s)\sim s\mu_D$$ as $$s\to 0^+$$ , we have $$0\le s\mu_D-(1-\varphi(s))\le 1-\varphi(s)$$ for sufficiently small s. Then we obtain

\[ \mathrm{e}^{-sX-\kappa_A (1-\varphi(s))} \kappa_A^2(s\mu_D-(1-\varphi(s)))^2\le \mathrm{e}^{-\kappa_A (1-\varphi(s))}\kappa_A^2(1-\varphi(s))^2=o(1), \]

where the negligibility follows from dominated convergence and the basic inequality $$\mathrm{e}^{-x} x^2\le 4\mathrm{e}^{-2}$$ for any $$x{\gt}0$$ . We obtain

\[ I_2=o\bigg(\frac{s\mu_D-(1-\varphi(s))}{s^2}\bigg)\quad\text{as } s\to 0^+. \]

A similar computation to above yields

\begin{equation*} 0\le \varphi(s)-\varphi_Y(s)\le {\mathbb{E}} [ \kappa_A (s\mu_D-(1-\varphi(s)) \mathrm{e}^{-sX+\kappa_A (\varphi(s)-1))}] \le \kappa(s\mu_D-(1-\varphi(s)). \end{equation*}

Thus, as $$(s\mu_D-(1-\varphi(s))\le (s\mu_D-(1-\varphi_Y(s))/(1-\kappa)$$ and $${\mathbb{E}}[Y]={\mathbb{E}}[X]+\kappa \mu_D=\mu_D$$ , we conclude that

\begin{equation*} \bigg|\varphi''(s)-\frac{\varphi_Y''(s)}{1-{\mathbb{E}}[\mathrm{e}^{-sX-\kappa_A (1-\varphi(s))}\kappa_A ] }\bigg| \\ =o\bigg(1+\frac{{\mathbb{E}}[Y] + \varphi_Y'(s)}s+\frac{s{\mathbb{E}}[Y]-(1-\varphi_Y(s))}{s^2}\bigg) \end{equation*}

as $$s\to 0^+$$ . Let us first apply Theorem 5.2 on Y so that $$\varphi''_Y(s)$$ is $$\alpha-2$$ regularly varying around 0. Applying Karamata’s theorem again, i.e. the equivalence between (8.1.11b) and (8.1.9) (8.1.11b) and (8.1.10) of Bingham et al. (Reference Bingham, Goldie and Teugels1987, Theorem 8.1.6) respectively assert that $$(s{\mathbb{E}}[Y]-(1-\varphi_Y(s)))/s^2=\smash{O(\varphi_Y''(s))}$$ and $$\smash{({\mathbb{E}}[Y] + \varphi_Y'(s))/s=O(\varphi_Y''(s))}$$ as $$s\to 0^+$$ . We then obtain

$$\varphi''(s)\sim \frac{\varphi''_Y(s)}{1-\kappa}\sim \frac{c s^{\alpha -2 }\ell(1/s)}{1-\kappa}\quad\text{as } s\to 0+, \$$

and, finally,

\begin{equation*} \skew4\bar F(x) \sim \frac{\skew4\bar F_Y(x)}{1-\kappa} \quad\text{as }x\to \infty, \end{equation*}

by applying the reverse part of Theorem 5.2.

We are now ready to characterize the asymptotic behavior of S(t) in the regularly varying case.

Theorem 5.3. Assume that the assumptions of Lemma 5.2 hold.

• (i) If $$\alpha \in (0,1)$$ and there exists $$\delta{\gt}0$$ such that

  (5.15) \begin{equation} t ^{ \delta }{\mathbb{E}} [\tilde{\mu}_A(t,\infty)] \to 0\quad\text{as }t {\to\infty}, \label{eqn34} \end{equation}
  then there exists a sequence $$(a_n),\, a_n{\to\infty},$$ and an $$\alpha$$ -stable random variable $$G_\alpha$$ such that
  \begin{equation*} \frac{S(t) }{ a_{\lfloor \nu t \rfloor} } \xrightarrow{\text{D}} G_\alpha. \end{equation*}

• (ii) If $$\alpha \in (1,2)$$ and

  (5.16) \begin{equation} t ^{1+\delta - 1/\alpha} {\mathbb{E}} [\tilde{\mu}_A(t,\infty)] \to 0\quad\text{as }t {\to\infty} \label{eqn35} \end{equation}
  holds for some $$\delta {\gt}0$$ , then there exists a sequence $$(a_n),\, a_n{\to\infty} ,$$ and an $$\alpha$$ -stable random variable $$G_\alpha$$ such that
  \begin{equation*} \frac{S(t) - t \nu \mu_D }{ a_{\lfloor \nu t \rfloor} } \xrightarrow{\text{D}} G_\alpha. \end{equation*}

Proof. The proof is based on representation (3.1), and application of Propositions 4.1 and 4.2. In either case, it remains to show that

\begin{equation*} {\varepsilon}_t = o_P(a_t). \end{equation*}

Consider first the case $$\alpha \in (1,2)$$ . Since then $$\mu_D = {\mathbb{E}} D{\lt}\infty$$ , the argument in the proof of Theorem (5.1) still yields bound (5.12) on $${\mathbb{E}} {\varepsilon}_t$$ . Using l’Hôpital’s rule again together with condition (5.16), we obtain $${\mathbb{E}} {\varepsilon}_t = o (t^{1/\alpha-\delta})$$ , where we assume without loss of generality that $$\delta{\lt}1/\alpha$$ . Since $$a_t = t^{1/\alpha} \ell(t)$$ for some slowly varying function $$\ell$$ , it follows that $${\varepsilon}_t /a_t\xrightarrow{{\mathbb{P}\ }} 0$$ as $$t{\to\infty}$$ .

For $$\alpha \in (0,1)$$ , the random variable D has no finite mean. In order to prove that $$ {\varepsilon}_t = o_P(a_t)$$ , we use the Markov inequality of order $$0{\lt}\gamma{\lt}\alpha$$ as $${\mathbb{E}} [D^\gamma]{\lt}\infty$$ . We will show that, under assumption (5.15),

\[ {\mathbb{E}} [{\varepsilon}_t^\gamma]=o(a_t^\gamma)\quad\text{as } t\to \infty. \]

By the sublinearity of $$x\to x^\gamma,\,\gamma\le1$$ , we have

\begin{align*} {\mathbb{E}}[{\varepsilon}_t^\gamma] &= {\mathbb{E}}\bigg[\bigg( \sum_{{\Gamma_i} \leq t}\sum_j {{\bf 1}}_{\{\Gamma_i + T_{ij} {\gt}t\}} X_{ij}\bigg)^\gamma\bigg] \\ &= {\mathbb{E}}\bigg[\bigg(\sum_{\tau_i \leq t} \sum_{\tau_j {\gt} t} D_{\tau_j} {{\bf 1}}_{\{\tau_i\to \tau_j\}}\bigg)^\gamma\bigg] \\ &\leq {\mathbb{E}}\bigg[ \sum_{\tau_i \leq t} \sum_{\tau_j {\gt} t} D_{\tau_j}^\gamma {{\bf 1}}_{\{\tau_i\to \tau_j\}} \bigg] \\ &={\mathbb{E}}\bigg[\sum_{\tau_i \leq t} {\mathbb{E}}\bigg[\sum_{k=1}^{L^{i}} D_{\tau_i+W_{ik}}^\gamma {{\bf 1}}_{\{\tau_i+W_{ik} {\gt} t\}} \biggm| (\tau_i,A^i)_{i\ge 0}; \,\tau_i \le t\bigg]\bigg] \\ &= {\mathbb{E}}[D^\gamma] {\mathbb{E}} \bigg[ \int_{0}^t \int_{\mathbb{S}} \tilde{\mu}_a((t-s,\infty)) N(\mathrm{d} s,\,\mathrm{d} a) \bigg]. \end{align*}

We can again compare the marked Hawkes process N with a stationary version of it, $$N^*$$ say. By the same arguments as in the proof of Theorem 5.1, we obtain

\begin{eqnarray*} {\mathbb{E}} \bigg[ \sum_{0\le \tau_i \leq t} \sum_{t{\lt} \tau_j} {{\bf 1}}_{\{\tau_i\to \tau_j\}} \bigg]\le \dfrac{\nu}{1-\kappa}\bigg(\int_0^t\int_u^\infty {\mathbb{E}}[h(s,A)]\,\mathrm{d} s\,\mathrm{d} u\bigg). \end{eqnarray*}

By regular variation of order $$1/\alpha$$ of $$(a_t)$$ we have $$t^{\gamma/\alpha-\delta'}=o(a_t^\gamma)$$ for any $$\delta'{\gt}0$$ . Once again we use a Cesaró argument to prove that $${\mathbb{E}} [{\varepsilon}_t^\gamma]=o(a_t^\gamma)$$ under the condition that

\[ t ^{1+\delta' - \gamma/\alpha}{\mathbb{E}} [\tilde{\mu}_A(t,\infty)] \to 0 \quad\text{as } t\to \infty. \]

As $$\gamma$$ can be taken as close as possible to $$\alpha$$ , the result holds under assumption (5.15).