2 Main results

2.1. Assumptions

Definition 2.1. Let $0 < \alpha \le 2$ . Write ${\varepsilon} \in \mathcal{D}(\alpha)$ if the distribution of a random variable ${\varepsilon}$ satisfies the following conditions:

1. • for $\alpha=2$ , ${\mathbb{E}}{\varepsilon}^2<\infty$ ;

2. • for $0 < \alpha < 2$ , there exist some finite constants $c_1,c_2 \ge 0$ such that

   \begin{equation*}\lim_{x\to\infty}x^{\alpha}{\mathbb{P}}({\varepsilon}>x)=c_1,\quad \lim_{x\to-\infty}|x|^{\alpha}{\mathbb{P}}({\varepsilon} \le x)=c_2,\quad c_1+c_2>0;\end{equation*}

3. • in addition to the above, ${\mathbb{E}} {\varepsilon} = 0$ for $1 < \alpha \le 2$ , and, for $\alpha =1$ , the distribution of ${\varepsilon}$ is symmetric.

Remark 2.1. Assumption ${\varepsilon} \in \mathcal{D}(\alpha)$ implies that ${\varepsilon}$ belongs to the domain of normal attraction of an $\alpha$ -stable distribution. That is, for a sequence $\{{\varepsilon}(t), \, t = 1, 2, \dots \}$ of independent copies of ${\varepsilon}$ ,

(2.1) \begin{equation}n^{- 1/\alpha}\sum_{t=1}^{[n\tau]}{\varepsilon}(t) \to_{\textnormal{fdd}} \zeta_\alpha(\tau),\end{equation}

where $\zeta_\alpha = \{\zeta_\alpha(\tau), \, \tau \ge 0 \} $ is an $\alpha$ -stable Lévy process having characteristic function (see [Reference Feller8 , pages 574–581])

(2.2) \begin{align}&{\mathbb{E}} \text{e}^{{\text{i}}\theta \zeta_\alpha(\tau)}=\text{e}^{- \tau |\theta|^{\alpha}\omega(\theta)},\quad \theta\in {\mathbb{R}}, \quad \text{with} \notag\\[3pt] &\omega(\theta)\coloneqq \begin{cases}\frac{\Gamma(2-\alpha)}{1-\alpha} \bigg( (c_1+c_2)\cos \bigg(\frac{\alpha \pi}{2} \bigg)-{\text{i}}(c_1-c_2) {sign}(\theta) \sin \bigg(\frac{\alpha \pi}{2} \bigg) \bigg) &\alpha\ne 1,2, \\[9pt](c_1+c_2)\frac{\pi}{2} &\alpha=1,\\[9pt]\frac{1}{2} {\mathbb{E}} {\varepsilon}^2 &\alpha=2.\end{cases}\end{align}

Furthermore, assumption ${\varepsilon} \in \mathcal{D}(\alpha)$ implies ${\mathbb{E}} |{\varepsilon}|^p < \infty$ for any $0 < p < \alpha$ .

In what follows, we assume that $\{{\varepsilon} (t), \, t \in {\mathbb{Z}} \}$ in (1.2) are independent copies of ${\varepsilon} \in \mathcal{D}(\alpha)$ for some $0 < \alpha \le 2$ . Moreover, we assume that a is an absolutely continuous random variable having density $\phi$ which is supported on [0, 1) and admits the representation

(2.3) \begin{equation}\phi(u) = \psi(u)(1-u)^{\beta-1}, \quad u\in [0,1),\end{equation}

for some $\beta >0$ and some integrable function $\psi (u)$ , $u \in [0,1)$ , having finite limit $\lim_{u\uparrow 1}\psi(u)\break =\!:\,\psi_1>0$ . The same assumption is made in [Reference Pilipauskaitė and Surgailis27], [Reference Puplinskaitė and Surgailis31], and other related works. Then there exists a unique stationary solution of (1.2) given by

(2.4) \begin{equation}X(t) = \sum_{s \le t} a^{t-s} {\varepsilon} (s), \quad t \in {\mathbb{Z}},\end{equation}

where the series on the right-hand side of (2.4) converges in $L^p$ for $0 < p < \alpha \min\!(\beta, 1)$ if $0 < \alpha < 2$ , and for $0 < p \le 2$ such that $p < 2\beta$ if $\alpha = 2$ . For almost every $a \in [0,1)$ , the series on the right-hand side of (2.4) converges conditionally a.s. and conditionally in $L^p$ for $0 < p < \alpha$ if $0 < \alpha < 2$ , and for $0 < p \le 2$ if $\alpha = 2$ . See [Reference Puplinskaitė and Surgailis30] for details.

2.2. Limiting processes

For $1 < \beta < \alpha \le 2$ , we define a stochastic process $\Lambda_{\alpha,\beta} = \{ \Lambda_{\alpha,\beta}(\tau),\, \tau \ge 0 \}$ by

(2.5) \begin{equation} \Lambda_{\alpha,\beta} (\tau) \coloneqq \int_{{\mathbb{R}}_+ \times {\mathbb{R}}} f_\tau (x,s) M (\text{d} x, \text{d} s),\end{equation}

where

\begin{equation*}f_\tau (x,s) \coloneqq \int_0^\tau \text{e}^{-x (t-s)} {\textbf{1}} (s \le t) \, \text{d} t, \quad \tau \ge 0, \ x > 0, \ s \in {\mathbb{R}},\end{equation*}

and $M (\text{d} x, \text{d} s)$ is an $\alpha$ -stable random measure on ${\mathbb{R}}_+ \times {\mathbb{R}}$ with a control measure $m (\text{d} x, \text{d} s) \coloneqq \psi_1 x^{\beta-1} \, \text{d} x \, \text{d} s $ such that ${\mathbb{E}} \text{e}^{ \text{i} \theta M (B) } = \text{e}^{-|\theta|^\alpha \omega(\theta) m (B)}$ , $\theta \in {\mathbb{R}}$ , for every Borel set $B \subset {\mathbb{R}}_+ \times {{\mathbb{R}}}$ with $m (B) < \infty$ and $\omega$ , $\psi_1 $ given in (2.2) and (2.3). The process $\Lambda_{\alpha,\beta} $ was introduced in [Reference Puplinskaitė and Surgailis31]. It is $\alpha$ -stable, H-self-similar with $H = 1 - (\beta-1)/\alpha \in (1/\alpha, 1)$ , has stationary dependent increments, and is related to the integrated superposition of Ornstein–Uhlenbeck processes discussed by Barndorff-Nielsen [Reference Barndorff-Nielsen1]. See also [Reference Grahovac, Leonenko and Taqqu11]. The joint characteristic function of $\Lambda_{\alpha,\beta}$ is given by

(2.6) \begin{equation} {\mathbb{E}} \exp\bigg\{{\text{i}}\sum_{j=1}^d \theta_j \Lambda_{\alpha,\beta} (\tau_j) \bigg\} =\exp \bigg\{ - \int_{{\mathbb{R}}_+ \times {\mathbb{R}}}\bigg|\sum_{j=1}^d \theta_j f_{\tau_j} (x,s) \bigg|^\alpha \omega\bigg(\!\sum_{j=1}^d \theta_j f_{\tau_j} (x,s) \bigg) m ({\text{d}}x, {\text{d}}s) \bigg\},\end{equation}

for $\theta_j \in {\mathbb{R}}$ , $\tau_j \ge 0$ , $j=1, \dots, d$ , $d\in {\mathbb{N}}$ . For $\alpha = 2$ , $\Lambda_{2,\beta}$ is a Gaussian process with mean zero and the autocovariance function

(2.7) \begin{align}{\mathbb{E}} \Lambda_{2,\beta} (\tau_1) \Lambda_{2,\beta} (\tau_2) &= {\mathbb{E}} {\varepsilon}^2\int_{{\mathbb{R}}_+ \times {\mathbb{R}}} f_{\tau_1} (x,s)\, f_{\tau_2} (x,s) m (\text{d} x, \text{d} s)\notag\\[3pt] &=\frac{\sigma^2_\beta}{2} (\tau_1^{2H} + \tau_2^{2H} - |\tau_1-\tau_2|^{2H}), \quad \tau_1, \tau_2 \ge 0.\end{align}

It follows that $\Lambda_{2,\beta}$ is a fractional Brownian motion with Hurst index $H = (3-\beta)/2$ and variance

\begin{equation*} {\mathbb{E}} \Lambda^2_{2,\beta} (1) =\!:\, \sigma_\beta^2 = \psi_1 \Gamma (\beta-1) {\mathbb{E}} {\varepsilon}^2 /(2-\beta)(3-\beta).\end{equation*}

Next, for $0< \lambda < 1$ , $0 < \alpha \le 2$ , $\beta >0$ , let $W_{\lambda,\alpha, \beta} >0$ be a $\lambda$ -stable random variable with Laplace transform

(2.8) \begin{equation} {\mathbb{E}} \text{e}^{ - \theta W_{\lambda,\alpha, \beta} } =\text{e}^{ - \kappa_{\lambda,\alpha, \beta} \theta^{\lambda}}, \quad \theta \ge 0,\end{equation}

where

\begin{equation*}\kappa_{\lambda,\alpha, \beta} \coloneqq \psi_1\int_0^\infty (1- \exp \{ - (\lambda \alpha/\beta)^{-1} x^{-\beta/ \lambda} \}) x^{\beta-1} \, \text{d} x= \frac{\psi_1 \Gamma(1-\lambda)}{(\lambda \alpha/\beta)^\lambda \beta} > 0.\end{equation*}

It is well known (see e.g. [Reference Zolotarev37, Theorem 2.6.1]) that the Laplace transform in (2.8) extends to all complex numbers $\theta \in {\mathbb{C}} $ with ${Re}\!(\theta) \ge 0$ . Assume that $W_{\lambda,\alpha,\beta}$ is independent of the Lévy process $\zeta_\alpha $ in (2.1). Define

(2.9) \begin{equation} \begin{aligned}V_{\alpha,\beta}&\coloneqq W^{1/\alpha}_{\beta,\alpha,\beta} \, \zeta_\alpha (1), & & 0 < \beta < 1, \\[3pt] \mathcal{W}_{\alpha,\beta}(\tau)&\coloneqq W^{1/\alpha}_{\beta/\alpha,\alpha,\beta} \, \zeta_\alpha (\tau), & &\tau \ge 0, \ 0< \beta < \alpha.\end{aligned}\end{equation}

Then, using (2.8), we obtain for any $\theta \in {\mathbb{R}}$ ,

(2.10) \begin{equation} \begin{aligned}{\mathbb{E}} \text{e}^{\text{i} \theta V_{\alpha,\beta}}&={\mathbb{E}} \text{e}^{- |\theta|^\alpha \omega(\theta) W_{\beta, \alpha, \beta}}= \exp \{ - \kappa_{\beta,\alpha, \beta} |\theta|^{\alpha \beta} (\omega(\theta))^\beta \}, & & 0< \beta < 1, \\{\mathbb{E}} \text{e}^{\text{i} \theta \mathcal{ W}_{\alpha,\beta}(\tau)}&={\mathbb{E}} \text{e}^{-\tau |\theta|^\alpha \omega(\theta) W_{\beta/\alpha, \alpha, \beta}}= \exp \{ - \kappa_{\beta/\alpha,\alpha, \beta} \tau^{\beta/\alpha} |\theta|^{\beta} (\omega(\theta))^{\beta/\alpha} \}, & & 0< \beta < \alpha,\end{aligned}\end{equation}

where

(2.11) \begin{equation}\kappa_{\beta,\alpha, \beta} = \frac{\psi_1}{\alpha^\beta \beta}\Gamma (1- \beta), \quad\kappa_{\beta/\alpha, \alpha, \beta} = \frac{\psi_1}{\beta} \Gamma \bigg(1 - \frac{\beta}{\alpha}\bigg).\end{equation}

From (2.10), it follows that random variables $V_{\alpha,\beta} $ and $\mathcal{W}_{\alpha,\beta}(\tau)$ are stable with respective stability indices $\alpha \beta < \alpha$ and $\beta < \alpha$ . In a similar way, it follows that $\mathcal{W}_{\alpha,\beta} = \{ \mathcal{W}_{\alpha,\beta}(\tau), \, \tau \ge 0\}$ has $\beta$ -stable finite-dimensional distributions. Following [Reference Samorodnitsky and Taqqu34, Section 3.8], we call the stochastic processes in (2.9) sub-stable. We note that $\mathcal{W}_{\alpha,\beta}$ enjoys the stationary increment and H-self-similarity with $H = 1/\alpha $ properties which it inherits from the Lévy process $\zeta_\alpha$ . For $\beta =1 < \alpha \le 2$ , introduce also an $\alpha$ -stable random variable $V_{\alpha,1} $ with a characteristic function

(2.12) \begin{equation}{\mathbb{E}} \text{e}^{\text{i} \theta V_{\alpha,1}} = \text{e}^{-(\psi_1/\alpha)|\theta|^\alpha \omega(\theta)}, \quad \theta \in {\mathbb{R}}.\end{equation}

Since $\lim_{\beta \uparrow 1} (1-\beta)\Gamma(1-\beta) =1, $ it follows that $(1-\beta)^{1/(\alpha \beta)} V_{\alpha,\beta} \to_{\textnormal{d}} V_{\alpha,1}$ as $\beta \uparrow 1$ . The above discontinuity of the distribution of $V_{\alpha,\beta}$ at $\beta =1 $ can be explained by the additional logarithmic normalization in (2.16) and (2.21).

Finally, for $0 < \beta < \alpha \le 2$ , we define a random process $\mathcal{Z}_{\alpha, \beta} = \{ \mathcal{Z}_{\alpha,\beta} (\tau), \, \tau \ge 0 \}$ through its joint characteristic function:

(2.13) \begin{align} &{\mathbb{E}} \exp \bigg\{ \text{i} \sum_{j=1}^d \theta_j \mathcal{Z}_{\alpha,\beta} (\tau_j) \bigg\}\notag\\&\quad= \exp \bigg\{ \psi_1 \int_{{\mathbb{R}}_+} \bigg(\exp \bigg\{\!- \int_{{\mathbb{R}}} \bigg| \sum_{j=1}^d \theta_j f_{\tau_j} (x,s) \bigg|^\alpha\omega \bigg( \sum_{j=1}^d \theta_j f_{\tau_j} (x,s) \bigg) \, \text{d} s \bigg\} -1 \bigg) x^{\beta-1} \, \text{d} x \bigg\},\end{align}

where $\theta_j \in {\mathbb{R}}$ , $\tau_j \ge 0$ , $j=1, \dots, d$ , $d \in {\mathbb{N}}$ and $f_\tau (x,s)$ is given in (2.5). A stochastic integral representation and various properties of $\mathcal{Z}_{\alpha,\beta}$ are discussed in Section 3.

2.3. Limit theorems

In Theorems 2.1 and 2.2, the process $S_{N,n} = \{ S_{N,n} (\tau), \, \tau \ge 0 \}$ is the joint aggregate in (1.5) of independent copies of the RCAR(1) process $X = \{ X(t), \, t \in {\mathbb{Z}} \}$ in (2.4) satisfying the above-stated assumptions for some $0 < \alpha \le 2$ , some $\beta > 0$ and some $\psi_1 > 0$ . Theorem 2.1 discusses iterated limits when $N \to \infty $ followed by $n \to \infty $ (limits (2.14)–(2.16)), or vice versa (limits (2.17), (2.18)). Let $\kappa_\alpha \coloneqq {\mathbb{E}} (1-a)^{-\alpha} $ when the last expectation exists.

Theorem 2.1. Let $0 < \beta < \max\!(\alpha,1)$ . Then

(2.14) \begin{align} \text{(fdd)} \lim_{n \to \infty} \lim_{N \to \infty} n^{-1+(\beta-1)/\alpha} N^{-1/\alpha} S_{N,n}(\tau)&= \Lambda_{\alpha,\beta}(\tau), & \quad &1 < \beta < \alpha, \end{align}

(2.15) \begin{align} \text{(fdd)} \lim_{n \to \infty} \lim_{N \to \infty} n^{-1} N^{-1/(\alpha \beta)} S_{N,n}(\tau)&= V_{\alpha,\beta} \, \tau, & \quad & 0 < \beta < 1, \end{align}

(2.16) \begin{align}\text{(fdd)}\lim_{n\to \infty} \lim_{N\to\infty} n^{-1} (N \log N)^{-1/\alpha} S_{N,n} (\tau)&= V_{\alpha,1} \tau, &\quad & 1 = \beta < \alpha\end{align}

and

(2.17) \begin{align}\text{(fdd)} \lim_{N \to \infty} \lim_{n \to \infty} N^{-1/\beta} n^{-1/\alpha} S_{N,n}(\tau)&= \mathcal{W}_{\alpha,\beta}(\tau), & \quad &0 < \beta < \alpha,\end{align}

(2.18) \begin{alignat}{3}\text{(fdd)} \lim_{N \to \infty} \lim_{n \to \infty} N^{-1/\alpha} n^{-1/\alpha} S_{N,n}(\tau)&= \kappa^{1/\alpha}_\alpha \zeta_\alpha (\tau), & \quad & 0 < \alpha < \beta < 1.\end{alignat}

The following theorem discusses joint limits of appropriately normalized $S_{N,n}$ under simultaneous increase of N, n. As noted in the Introduction, these limits depend on the mutual increase rate of N, n and the parameters $\alpha, \beta $ . In (2.26) below, $V_{\alpha,\beta}$ and $\zeta_\alpha $ are mutually independent.

Theorem 2.2.

1. (i) Let $0 < \beta < \alpha$ . Let $N,n \to \infty$ so that

   \begin{equation*}\frac{N^{1/\beta}}{n} \to \mu \in [0,\infty].\end{equation*}
   Then
   (2.19) \begin{align} N^{-1/\alpha} n^{-1 + (\beta-1)/\alpha} S_{N,n}(\tau) \to_{\textnormal{fdd}}\Lambda_{\alpha,\beta} (\tau), \qquad\quad\,\,\,\,\,\, \mu = \infty, \ 1 < \beta < \alpha, \end{align}
   (2.20) \begin{alignat}{3} N^{-1/(\alpha \beta)} n^{-1} S_{N,n}(\tau) \to_{\textnormal{fdd}} V_{\alpha,\beta} \, \tau, \qquad\qquad\ \ \,\, \mu = \infty, \ 0 < \beta < \min\!(\alpha,1), \end{alignat}
   (2.21) \begin{align}(N \log (N/n))^{-1/\alpha} n^{-1} S_{N,n} (\tau) & \to_{\textnormal{fdd}} V_{\alpha,1} \tau, \qquad\qquad\quad\, \mu = \infty, \ 1 = \beta < \alpha, \end{align}
   (2.22) \begin{align}N^{-1/\beta} n^{-1/\alpha} S_{N,n}(\tau)&\to_{\textnormal{fdd}} \mathcal{W}_{\alpha,\beta}(\tau), \qquad\quad\ \ \ \ \mu = 0, \ 0 < \beta < \alpha, \end{align}
   (2.23) \begin{alignat}{3} N^{-1/\beta} n^{-1/\alpha} S_{N,n}(\tau)&\to_{\textnormal{fdd}} \mu^{1/\alpha}\mathcal{Z}_{\alpha,\beta}(\tau/\mu), \quad \mu \in (0,\infty),\ 0 < \beta < \alpha. \end{alignat}

2. (ii) Let $0 < \alpha < \beta < 1$ . Let $N,n \to \infty$ so that

   \begin{equation*}\frac{ N^{1/(\gamma\beta)}}{n} \to \mu \in [0,\infty], \quad \text{where } \gamma \coloneqq \frac{1-\alpha}{1-\beta} > 1.\end{equation*}
   Then
   (2.24) \begin{align}N^{-1/(\alpha \beta)} n^{-1} S_{N,n} (\tau) \to_{\textnormal{fdd}} V_{\alpha,\beta} \, \tau, \qquad\qquad\qquad\qquad\quad\ \ \, \mu = \infty, \end{align}
   (2.25) \begin{align}(Nn)^{-1/\alpha} S_{N,n} (\tau) \to_{\textnormal{fdd}} \kappa^{1/\alpha}_\alpha \zeta_\alpha (\tau), \qquad\qquad\qquad\qquad\, \mu = 0, \end{align}
   (2.26) \begin{align}(Nn)^{-1/\alpha} S_{N,n} (\tau) \to_{\textnormal{fdd}}\mu^{(1/\alpha)-1} V_{\alpha,\beta} \,\tau + \kappa^{1/\alpha}_\alpha \zeta_\alpha(\tau), \quad \mu \in (0,\infty). \end{align}

3. (iii) Let $\beta > \max\!(\alpha,1)$ . Then, as $N, n \to \infty$ in an arbitrary way,

   (2.27) \begin{equation}(Nn)^{-1/\alpha} S_{N,n} (\tau) \to_{\textnormal{fdd}} \kappa^{1/\alpha}_\alpha \zeta_\alpha (\tau).\end{equation}

Remark 2.2. We expect that results in Theorems 2.1 and 2.2, as well as in [Reference Pilipauskaitė and Surgailis27], can be extended to higher-order RCAR models, making suitable assumptions about the mixing distribution. In particular, Oppenheim and Viano [Reference Oppenheim and Viano25] discussed long memory properties of an RCAR(2p), $p\ge 1$ , model with autoregressive polynomial having one positive, one negative and $p-1$ pairs of non-real (complex conjugate) roots whose moduli are assumed to be independent random variables whose densities have power-law behavior at 1, similar to (1.3), for possibly different exponents $\beta_{i}$ . As shown in [Reference Oppenheim and Viano25], these assumptions lead to oscillating asymptotics, thus seasonal behavior of the autocovariance function of the RCAR process. A more forthright higher-order version of the RCAR(1) equation in (1.2) is the RCAR(p) model with real positive roots, namely,

(2.28) \begin{equation} (1- a_1 B) \cdots (1-a_p B) X(t) = {\varepsilon}(t), \quad t \in {\mathbb{Z}},\end{equation}

where $a_i \in [0,1)$ , $i=1, \dots, p$ , are independent random variables and $B X(t) = X(t-1)$ is the backward shift. The stationary solution of (2.28) is written as a moving average process $X(t) = \sum_{s\le t} b(t-s) {\varepsilon}(s)$ , $t\in{\mathbb{Z}}$ , where $b(t) \coloneqq \sum_{0 \le s_1 \le \dots \le s_{p-1} \le t}a_1^{s_1} a_2^{s_2-s_1} \cdots a_p^{t-s_{p-1}} $ satisfy

\begin{equation*} \sum_{t=0}^\infty b(t) = \prod_{i=1}^p (1-a_i)^{-1} =\!:\, A.\end{equation*}

In particular, it follows that given $a_1, \dots, a_p$ , conditionally

(2.29) \begin{equation}n^{-1/2} \sum_{t=1}^{[nt]} X(t) \to_{\textnormal{fdd}} A B(\tau),\end{equation}

which agrees with (4.9) below for $p=1$ , $\alpha =2$ . In the case when each $a_i$ has a density satisfying a similar relation to (1.3) for some $\beta_i>0$ , $\psi_{1i}>0$ , $i=1,\dots,p$ , the (random) factor A in (2.29) has a heavy-tailed distribution with tail parameter $\beta_{\min} \coloneqq \min_{1\le i \le p} \beta_i$ (see [7, Corollary, page 245]), and we can expect that, for $\beta_{\min} < \alpha =2$ , the suitably normalized iterated limit ${(fdd)} \lim_{N \to \infty} \lim_{n \to \infty}S_{N,n}(\tau)$ is the sub-Gaussian process $\mathcal{W}_{2,{\beta_{\min}}}(\tau)$ , following the proofs of Theorem 2.1 (2.17) or [Reference Pilipauskaitė and Surgailis27, Theorem 2.1 (2.10)]. We also then expect that the suitably normalized iterated limit ${(fdd)} \lim_{n \to \infty} \lim_{N \to \infty} S_{N,n}(\tau)$ is a fractional Brownian motion with Hurst parameter $H = (3 - \beta_{\min})/2$ (or a stable self-similar process in the case when X(t) has infinite variance). A challenging open problem is to make the above argument rigorous and to extend it to joint limits of $S_{N,n}$ as in Theorem 2.2.

3. The intermediate process

This section discusses properties of the intermediate process $\mathcal{Z}_{\alpha, \beta}$ introduced in (2.13) via its finite-dimensional characteristic function. We study Poisson stochastic integral representation, local and global self-similarity, a.s. continuity, and other properties of $\mathcal{Z}_{\alpha, \beta}$ . The results extend [Reference Pilipauskaitė and Surgailis27, Proposition 3.1] from $\alpha =2$ to $0< \alpha < 2$ . Roughly speaking, the Poisson integral representation of $\mathcal{Z}_{\alpha, \beta}$ is obtained by replacing the Brownian motion in [Reference Pilipauskaitė and Surgailis27] by Lévy process $\zeta_\alpha$ . However, some properties of $\mathcal{Z}_{\alpha, \beta}$ are not ‘continuous’ at $\alpha = 2$ . In particular, the second moment of $\mathcal{Z}_{\alpha, \beta}$ does not exist for $\alpha < 2$ , while $\mathcal{Z}_{2, \beta}$ may have higher moments than 2; see [Reference Pilipauskaitė and Surgailis27]. Clearly, these moment differences between the cases $\alpha < 2$ and $\alpha =2$ are related to the differences between the $\alpha$ -stable Lévy process $\zeta_\alpha$ , $\alpha < 2$ , and the Brownian motion $\zeta_2 = B$ .

Assume that the homogeneous Lévy process $\zeta_\alpha $ in (2.1) is extended to the whole real line ${\mathbb{R}} $ and induces a probability measure ${\mathbb{P}}_\alpha $ on the Borel sets of the Skorokhod space $D({\mathbb{R}})$ of càdlàg functions from ${\mathbb{R}} $ to ${\mathbb{R}}$ . We start with a family

(3.1) \begin{equation}z (\tau;\ x) \coloneqq \int_{{\mathbb{R}}} f_\tau (x,s) \, \text{d} \zeta_\alpha (s), \quad \tau \ge 0, \ x > 0,\end{equation}

of integrated Ornstein–Uhlenbeck processes driven by $\zeta_\alpha$ , where $f_\tau (x,s)$ is defined in (2.5). The process $\mathcal{Z}_{\alpha, \beta}$ can be defined by ‘mixing’ the above elementary processes of (3.1) on the path space $D({\mathbb{R}})$ of the Lévy process as follows.

Let $N (\text{d} x, \text{d} \zeta)$ denote a Poisson random measure on the product space ${\mathbb{R}}_+ \times D({\mathbb{R}})$ with mean $\nu (\text{d} x, \text{d} \zeta) = \psi_1 x^{\beta-1} \, \text{d} x \times {\mathbb{P}}_\alpha (\text{d} \zeta )$ , where $\psi_1 > 0$ , $0 < \beta < \alpha \le 2$ . Then $\mathcal{Z}_{\alpha, \beta} = \{ \mathcal{Z}_{\alpha, \beta} (\tau), \, \tau \ge 0 \}$ can be defined as a stochastic integral with respect to the above Poisson measure:

(3.2) \begin{align}\mathcal{Z}_{\alpha,\beta} (\tau) &\coloneqq \int_{(0,1) \times D({\mathbb{R}})} z (\tau;\ x ) N(\text{d} x, \text{d} \zeta) \notag \\[3pt] &\quad\,+\int_{[1,\infty) \times D({\mathbb{R}})} z (\tau;\ x ) ( N(\text{d} x, \text{d} \zeta) - \nu (\text{d} x, \text{d} \zeta) {\textbf{1}} (\alpha > 1) ).\end{align}

If $1 < \alpha \le 2$ , $1/\alpha < \beta < \alpha$ , then the two integrals in (3.2) can be combined into a single one:

(3.3) \begin{equation}\mathcal{Z}_{\alpha,\beta} (\tau) = \int_{{\mathbb{R}}_+ \times D({\mathbb{R}})} z (\tau;\ x) ( N (\text{d} x, \text{d} \zeta) - \nu (\text{d} x, \text{d} \zeta) ).\end{equation}

These and other properties of $\mathcal{Z}_{\alpha,\beta}$ are stated in the following proposition (we refer to [Reference Pilipauskaitė and Surgailis27] and [Reference Rajput and Rosinski32] for general properties of stochastic integrals with respect to Poisson random measure).

Remark 3.2. If $\mathcal{Z}_1, \mathcal{Z}_2, \dots, \mathcal{Z}_N$ are independent copies of $\mathcal{Z} \coloneqq \mathcal{Z}_{\alpha,\beta}$ , then, for any $N \in {\mathbb{N}}$ ,

(3.4) \begin{equation}\mathcal{Z} (\tau/N^{1/\beta}) ={} _{\textnormal{fdd}} N^{-1/(\alpha\beta)-1/\beta} \sum_{i=1}^N \mathcal{Z}_{i}(\tau).\end{equation}

Relation (3.4) follows from infinite divisibility of Poisson random measure $N(\text{d} x, \text{d} \zeta)$ in the stochastic integral representation (3.2) or the characteristic function (2.13). See also [Reference Pilipauskaitė and Surgailis27, (3.30)], where the above property is related to the aggregate-similarity property introduced by Kaj [Reference Kaj15]. For $0<\beta<\min\!(\alpha,1)$ , (3.4) and Proposition 3.1(v) imply that $N^{1/\beta} \mathcal{Z} (\tau/N^{1/\beta}) =_{\textnormal{d}} N^{-1/(\alpha\beta)} \sum_{i=1}^N \mathcal{Z}_i (\tau) \to_{\textnormal{d}} V_{\alpha,\beta} \, \tau$ as $N\to\infty$ . It follows that for a fixed $\tau >0$ , the (marginal) distribution of $\mathcal{Z} (\tau) \equiv \mathcal{Z}_{\alpha,\beta}(\tau)$ belongs to the domain of normal attraction of an $(\alpha\beta)$ -stable distribution, that is, $\mathcal{Z}_{\alpha,\beta}(\tau) \in \mathcal{D}(\alpha\beta)$ except possibly for the case $\alpha\beta = 1$ , when the distribution of $\mathcal{Z}_{\alpha,\beta}(\tau)$ is not symmetric. Similarly,

\begin{equation*}N^{H/\beta} \mathcal{Z}(\tau/N^{1/\beta}) ={}_{\textnormal{d}} N^{-1/\alpha} \sum_{i=1}^N \mathcal{Z}_i (\tau) \to_{\textnormal{d}} \Lambda_{\alpha,\beta} (\tau),\end{equation*}

where $H=1-(\beta-1)/\alpha$ , implying $\mathcal{Z}_{\alpha,\beta} (\tau) \in \mathcal{D}(\alpha)$ for $1<\beta<\alpha$ . These facts entail the precise asymptotic behavior of tail probabilities of $\mathcal{Z}_{\alpha,\beta} (\tau)$ for $\alpha < 2 $ and $\tau >0$ fixed; in particular, they show that condition $p < \alpha \min\!(\beta,1)$ in Proposition 3.1(ii) cannot be improved.

Remark 3.3. Let $0 < \alpha < \beta < 1$ . The limit process $\{ \mathcal{Z}^*_{\alpha,\beta} (\tau) \coloneqq V_{\alpha,\beta}\, \tau+ \kappa^{1/\alpha}_\alpha \zeta_\alpha (\tau), \, \tau \ge 0 \}$ in (2.26), Theorem 2.2(ii) can be also regarded as a ‘bridge’ between the other two limit processes in (2.24) and (2.25) since it is both locally and globally asymptotically self-similar:

\begin{alignat*}{3}c^{-1} \mathcal{Z}^*_{\alpha,\beta} (c \tau) &\to_{\textnormal{fdd}} V_{\alpha,\beta}\, \tau, & \quad & \text{as } c \to 0,\\[3pt] c^{-1/\alpha} \mathcal{Z}^*_{\alpha,\beta}(c \tau) &\to_{\textnormal{fdd}} \kappa^{1/\alpha}_\alpha \zeta_\alpha (\tau), & \quad & \text{as } c \to \infty.\end{alignat*}

4. Proofs

We first present some preliminary facts that will be used in the proofs.

Let $0 < \alpha \le 2$ . The characteristic function of a random variable ${\varepsilon} \in \mathcal{D}(\alpha)$ has the following representation in a neighborhood of the origin (see e.g. [Reference Ibragimov and Linnik13, Theorem 2.6.5]): there exists an $\epsilon>0$ such that

(4.1) \begin{equation}{\mathbb{E}} {\text{e}}^{{\text{i}} \theta {\varepsilon}} = {\text{e}}^{-|\theta|^\alpha \omega(\theta) h(\theta)} \quad \text{for any} \ \theta \in {\mathbb{R}}, \ |\theta| < \epsilon,\end{equation}

where $h (\theta)$ is a positive function tending to 1 as $\theta \to 0$ and $\omega(\theta)= \omega ( {sign}(\theta) )$ is the same as in (2.2).

For $a\in [0,1)$ and $n\in {\mathbb{N}}$ , let $c_n (a,s) \coloneqq \sum_{t=1}^n a^{t-s} {\textbf{1}} (s \le t)$ and note the following elementary inequalities:

(4.2) \begin{equation} \begin{split}\sum_{s \le 0} |c_n(a,s)|^\alpha& \le \frac{1}{\min\!(\alpha,1) (1-a)} \min \bigg(n, \frac{1}{1-a} \bigg)^\alpha,\\[3pt] \sum_{s=1}^n |c_n (a,s)|^\alpha & \le n \min \bigg(n, \frac{1}{1-a} \bigg)^\alpha.\end{split}\end{equation}

For $z \in \mathbb{C}$ , ${Re}(z) \le 0$ , we have

(4.3) \begin{equation}| \text{e}^{z} - 1 | \le \min\!(2, |z|), \quad |\text{e}^{z} - 1 - z| \le |z|^2.\end{equation}

Proof of Theorem 2.1. The iterated limits (2.14), (2.15) follow from [Reference Puplinskaitė and Surgailis31, Theorems 2.1, 3.1, Proposition 2.3].

Proof of (2.16). It suffices to prove that

\begin{equation*}\bar X_N (t) \coloneqq (N\log N)^{-1/\alpha} \sum_{i=1}^N X_i (t) \to_{\textnormal{fdd}} V_{\alpha,1},\end{equation*}

or that, for any $d \in {\mathbb{N}}$ and $\theta_t \in {\mathbb{R}}$ , $t=1,\dots,d$ ,

(4.4) \begin{equation}{\mathbb{E}} \text{e}^{ \text{i} \sum_{t=1}^d \theta_t \bar X_N (t)} \to {\mathbb{E}} \text{e}^{\text{i} \sum_{t=1}^d \theta_t V_{\alpha,1}} = \text{e}^\Theta \quad \text{with } \Theta \coloneqq - \frac{\psi_1}{\alpha} \bigg| \sum_{t=1}^d \theta_t \bigg|^\alpha \omega \bigg( \sum_{t=1}^d \theta_t \bigg).\end{equation}

Since $X_1,\dots,X_N$ are independent copies of X in (2.4), the left-hand side of (4.4) can be rewritten as $( 1+\gfrac{\Theta_{N}}{N} )^N$ with

(4.5) \begin{align}\Theta_{N}&\coloneqq N {\mathbb{E}} \bigg[ \exp \bigg\{ \text{i} (N \log N)^{-1/\alpha} \sum_{t=1}^d \theta_t X(t) \bigg\} - 1 \bigg]\notag \\[3pt]&= N \int_{[0,1)} \bigg( \prod_{s \in {\mathbb{Z}}} {\mathbb{E}} \exp \bigg\{ \text{i} (N \log N)^{-1/\alpha} c (u,s) {\varepsilon} (s) \bigg\} - 1 \bigg) \phi(u) \, \text{d} u \notag \\[3pt] &= N \int_{[0,1)} \bigg( \exp\bigg\{ - \frac{K_N(u)}{ (1-u) N \log N }\bigg\} - 1 \bigg) \phi(u) \, \text{d} u,\end{align}

where

\begin{align*}c(u,s) &\coloneqq \sum_{t=1}^d \theta_t u^{t-s} {\textbf{1}} (s \le t),\\*K_N(u) &\coloneqq (1-u) \sum_{s \in {\mathbb{Z}}} | c(u,s) |^\alpha \omega ( c (u,s) ) h ( (N\log N)^{-1/\alpha}c (u,s) )\end{align*}

(in (4.5) we used (4.1) and the fact that c(u,s) is bounded uniformly in $u \in [0,1), s \in {\mathbb{Z}}$ ). For $\delta \in (0,1) $ split

\begin{align*}\Theta_N&= N \bigg\{\int_{0}^{1-\delta} + \int_{1-\gfrac{\delta}{N}}^1 + \int_{1-\delta}^{1-\gfrac{\delta}{N}} \bigg\} \bigg(g \exp\bigg\{ - \frac{K_N(u)}{ (1-u) N \log N } \bigg\} - 1 \bigg)\phi(u) \, \text{d} u\\*&=\!:\, \sum_{i=1}^3 \Theta_{N,\delta}^{i}.\end{align*}

Then (4.4) or $\lim_{N \to \infty} \Theta_N = \Theta $ follows from

(4.6) \begin{equation}\lim_{\delta \to 0} \limsup_{N \to \infty}|\Theta_{N,\delta}^{i}| =0, \ i=1,2, \quad \lim_{\delta \to 0} \limsup_{N \to \infty}|\Theta_{N,\delta}^3- \Theta| =0.\end{equation}

Here,

\begin{equation*}| \Theta_{N,\delta}^2 |\le 2 N \int_{1-\gfrac{\delta}{N}}^1 \phi (u) \, \text{d} u \le C \delta\end{equation*}

for all N large enough, implying (4.6) for $i=2 $ . Next, using (4.2),

\begin{equation*}\sum_{s \in {\mathbb{Z}}} |c(u,s)|^\alpha\le C (1-u)^{-1}.\end{equation*}

Therefore, $|K_N(u)| \le C$ , $u \in [0,1)$ , and by (4.3) we obtain

\begin{equation*} | \Theta_{N, \delta}^1 | \le C (\log N)^{-1} \int_0^{1-\delta} (1-u)^{-1} \phi(u) \, \text{d} u= C (\delta \log N)^{-1}, \end{equation*}

proving (4.6) for $i=1$ .

Consider the last relation in (4.6). In view of (2.3), we can replace $\Theta_{N,\delta}^3$ with

\begin{align*}\Theta_{N,\delta}^4 & \coloneqq \psi_1 N \int_{ 1-\delta}^{1-\gfrac{\delta}{N}} \bigg( \exp \bigg\{ - \frac{K_N(u)}{(1-u)N} \frac{1}{\log N} \bigg\} - 1 \bigg) \, \text{d} u \\[3pt] & = \psi_1 \int_{\delta}^{\delta N} \bigg( \exp \bigg\{ - \frac{K_N ( 1-\gfrac{x}{N})}{x \log N} \bigg\} - 1 \bigg) \, \text{d} x,\end{align*}

which in turn can be replaced by

\begin{equation*}\Theta_{N,\delta}^5 \coloneqq - \psi_1 \int_\delta^{\delta N} \frac{K_N (1-\gfrac{x}{N})}{x \log N} \, \text{d} x\end{equation*}

since

\begin{equation*}|\Theta_{N,\delta}^4 - {\Theta}^{5}_{N,\delta} | \le C \int_\delta^{\delta N} \frac{\text{d} x}{(x \log N)^{2}} = {\mathrm{o}} (1),\quad N \to \infty,\end{equation*}

follows from (4.3). We can rewrite $\Theta $ in (4.4) in a similar way:

\begin{equation*}\Theta = - \psi_1 K \int_{\delta}^{\delta N} \frac{\text{d} x}{x \log N} \quad \text{with } K \coloneqq \frac{1}{\alpha} \bigg| \sum_{t=1}^d \theta_t \bigg|^\alpha \omega \bigg( \sum_{t=1}^d \theta_t \bigg).\end{equation*}

Thus, the last relation in (4.6) follows from

\begin{equation*}\lim_{\delta \to 0} \limsup_{N \to \infty} \int_\delta^{\delta N} \bigg|K_N \bigg(1-\frac{x}{N}\bigg)- K \bigg| \frac{\text{d} x}{x \log N}= 0\end{equation*}

or

(4.7) \begin{equation}\limsup_{N \to \infty} \sup_{\delta < x < \delta N} \bigg|K_N \bigg(1-\frac{x}{N}\bigg)- K\bigg| \le \epsilon(\delta),\end{equation}

where $\lim_{\delta \to 0} \epsilon(\delta) = 0$ . To prove (4.7), denote $|z|_\omega^\alpha \coloneqq |z|^\alpha \omega (z)$ , $z \in {\mathbb{R}},$ and note that

\begin{equation*}\sup_{0 < x < \delta N} \bigg|K_N \bigg(1-\frac{x}{N}\bigg) -\tilde K\bigg(\frac{N}{x}\bigg)\bigg| = {\mathrm{o}} (1),\quad N \to \infty,\end{equation*}

where

(4.8) \begin{align}\tilde K(y)&\coloneqq \frac{1}{y} \sum_{s\in {\mathbb{Z}}} \bigg|\sum_{t=1}^d \theta_t \bigg( 1 - \frac{1}{y} \bigg)^{t-s} {\textbf{1}}(s\le t)\bigg|^\alpha_\omega \notag \\[3pt]&=\int_{{\mathbb{R}}} \bigg|\sum_{t=1}^d \theta_t \bigg(1 - \frac{1}{y} \bigg)^{t-[sy]} {\textbf{1}}([sy]\le t)\bigg|^\alpha_\omega \, \text{d} s\notag \\[3pt] &\to\int_{-\infty}^0 \bigg|\sum_{t=1}^d \theta_t \text{e}^{s} \bigg|^\alpha_\omega \, \text{d} s \notag \\[3pt] &= K, \quad y \to \infty,\end{align}

by the dominated convergence theorem (DCT) using $1 - z \le \text{e}^{-z}$ , $z \ge 0$ . Hence,

\begin{equation*}\sup_{0 < x < \delta N} \bigg|\tilde K\bigg(\frac{N}{x}\bigg) - K\bigg| \le \epsilon (\delta) = {\mathrm{o}} (1),\quad \delta \to 0,\end{equation*}

implying (4.7) and (4.6). This completes the proof of (2.16).

Proof of (2.17). Let us first prove that

(4.9) \begin{equation}n^{-1/\alpha} S_{n} (\tau) \coloneqq n^{-1/\alpha} \sum_{t=1}^{[n\tau]} X (t) \to_{\textnormal{fdd}} (1-a)^{-1} \zeta_\alpha (\tau),\end{equation}

where $\zeta_\alpha$ , X are the same as in (2.1), (2.4), and $a \in [0,1)$ is fixed. It suffices to show that, for any $d \in {\mathbb{N}}$ and $\tau_j > 0$ , $\theta_j \in {\mathbb{R}}$ , $j = 1, \dots, d$ ,

(4.10) \begin{equation}{\mathbb{E}}_a \exp \bigg\{ \text{i} n^{-1/\alpha} \sum_{j=1}^d \theta_j S_{n} (\tau_j) \bigg\} \to{\mathbb{E}}_a \exp \bigg\{ \text{i} (1-a)^{-1} \sum_{j=1}^d \theta_j \zeta_\alpha (\tau_j) \bigg\},\end{equation}

where ${\mathbb{E}}_a [\cdot] = {\mathbb{E}} [\cdot |a ]$ stands for conditional expectation. For brevity of notation, we restrict the proof of (4.10) (as well as all the rest in this theorem) to $d = 1$ and $\tau_1 = \tau > 0$ , $\theta_1 = \theta \in {\mathbb{R}}$ . Split

\begin{equation*} {\mathbb{E}}_a [\text{e}^{\text{i} \theta n^{-1/\alpha} S_n (\tau)} ( {\textbf{1}} (a \in I_n) + {\textbf{1}} (a \in I_n^c) ) ] =\!:\, \Phi'_n (\theta,a) + \Phi''_{\kern-5pt n}\, (\theta,a), \end{equation*}

where

\begin{equation*} I_n \coloneqq \bigg[0,1 - \frac{\log n}{n^{1/\alpha}} \bigg),\quad I_n^c \coloneqq [0,1) \setminus I_n .\end{equation*}

For $c_n(a,s)$ in (4.2),

\begin{equation*}\sup_{a \in I_{n}, s \in {\mathbb{Z}}} | n^{-1/\alpha} c_{[n\tau]}(a,s) | = {\mathrm{O}} ( (\log n)^{-1} ) = {\mathrm{o}} (1).\end{equation*}

Hence, for all n large enough, we can use (4.1) to rewrite $\Phi^{\prime}_{\kern-1pt n}(\theta,a)$ as

\begin{equation*}\Phi^{\prime}_{\kern-1pt n} (\theta,a) = \text{e}^{ - |\theta|^\alpha \omega (\theta) K_n (a) } {\textbf{1}} (a \in I_n),\end{equation*}

where

\begin{equation*}K_n (a) \coloneqq n^{-1} \sum_{s \in {\mathbb{Z}}} |c_{[n\tau]}(a,s) |^\alpha h ( \theta n^{-1/\alpha} c_{[n\tau]}(a,s) )\end{equation*}

and $ \sup_{a \in I_n, s \in {\mathbb{Z}}} | h ( \theta n^{-1/\alpha} c_{[n\tau]}(a,s) ) - 1 | = {\mathrm{o}} (1)$ . Then (4.10) follows from $\lim_{n \to \infty} K_n (a) =\tau (1- a)^{-\alpha}$ for every $a \in [0,1)$ and $\lim_{n \to \infty} \Phi''_{\kern-5.5pt n}\kern0.2pt (\theta,a) = 0$ . Both these facts are completely elementary, and we omit the details. This proves (4.9) for $d = 1$ . The proof for $d > 1$ follows similarly.

Let $\{ (1-a_i)^{-1} \zeta_{\alpha, i} (\tau), \, \tau \ge 0 \}$ , $i=1,2, \dots$ , be independent copies of $\{ (1-a)^{-1} \zeta_\alpha (\tau), \, \tau \ge 0 \}$ . With (4.9) in mind, (2.17) follows from

(4.11) \begin{equation}N^{-1/\beta} \sum_{i=1}^N (1-a_i)^{-1} \zeta_{\alpha,i} (\tau) \to_{\textnormal{fdd}} \mathcal{W}_{\alpha,\beta} (\tau).\end{equation}

Consider the one-dimensional convergence in (4.11) at $\tau = 1$ . For $\theta \in {\mathbb{R}}$ , we have

\begin{equation*}{\mathbb{E}} \exp \bigg\{ \text{i} \theta N^{-1/\beta}\sum_{i=1}^N (1-a_i)^{-1} \zeta_{\alpha,i} (1)\bigg\} =({\mathbb{E}} \exp \{ \text{i} \theta N^{-1/\beta} (1-a)^{-1} \zeta_\alpha (1) \} )^N = \bigg(1 + \frac{\Theta_N}{N} \bigg)^N,\end{equation*}

where

\begin{equation*}\Theta_N \coloneqq N{\mathbb{E}} [\text{e}^{-N^{-\alpha/\beta} (1-a)^{-\alpha} |\theta|^\alpha \omega (\theta)} - 1].\end{equation*}

We also have ${\mathbb{E}} \text{e}^{ \text{i} \theta \mathcal{W}_{\alpha,\beta} (1)} = \text{e}^{- \kappa_{\beta/\alpha,\alpha, \beta} |\theta|^\beta (\omega(\theta))^{\beta/\alpha}}$ ; see (2.10). The desired convergence in (4.11) at $\tau = 1$ follows from

(4.12) \begin{equation}\Theta_N \to - \kappa_{\beta/\alpha,\alpha, \beta} |\theta|^\beta (\omega(\theta))^{\beta/\alpha} \quad \text{{for all} } \theta \in {\mathbb{R}}.\end{equation}

By assumption (1.3), there exists an $\epsilon > 0$ and a constant $C > 0$ such that $\phi (u) \le C (1-u)^{\beta-1}$ for all $u \in [1-\epsilon,1)$ . Split

\begin{align*}\Theta_N &= N{\mathbb{E}}[ (\text{e}^{- N^{-\alpha/\beta} (1-a)^{-\alpha} |\theta|^\alpha \omega (\theta)} - 1 ) ( {\textbf{1}} (0\le a < 1-\epsilon) + {\textbf{1}} (1-\epsilon \le a < 1) ) ]\\[4pt]&=\!:\,\Theta^0_N + \Theta^1_N,\end{align*}

where

\begin{equation*}|\Theta^0_N| \le N^{1-\alpha/\beta} C \epsilon^{-\alpha} |\theta|^\alpha {\mathbb{P}} ( 0 \le a < 1- \epsilon ) = {\mathrm{o}} (1)\end{equation*}

since $\beta < \alpha.$ Hence, it suffices to prove (4.12) for $\Theta^1_N$ instead of $\Theta_N$ . By a change of variable,

\begin{align*}\Theta^1_N &= N \int_{1-\epsilon}^1 (\text{e}^{-N^{-\alpha/\beta} (1-u)^{-\alpha} |\theta|^\alpha \omega (\theta) } - 1 ) \phi (u) \, \text{d} u\\*&= \int_0^{\epsilon N^{1/\beta}} (\text{e}^{-x^{-\alpha} |\theta|^{\alpha} \omega (\theta) } - 1 ) N^{1-1/\beta} \phi \bigg( 1 - \frac{x}{N^{1/\beta}} \bigg) \, \text{d} x,\end{align*}

where

\begin{equation*}N^{1-1/\beta} \phi \bigg( 1 - \frac{x}{N^{1/\beta}} \bigg) \to \psi_1 x^{\beta-1}\quad\text{for } x \in {\mathbb{R}}_+\end{equation*}

by assumption (1.3). Moreover,

\begin{equation*}N^{1-1/\beta} \phi \bigg( 1 - \frac{x}{N^{1/\beta}} \bigg) \le C x^{\beta-1}\quad \text{for } x \in (0, \epsilon N^{1/\beta}].\end{equation*}

Therefore, the DCT implies that

\begin{equation*}\Theta_N^1 \to \psi_1 \int_0^\infty (\text{e}^{-x^{-\alpha} |\theta|^{\alpha} \omega (\theta) } - 1 ) x^{\beta-1} \, \text{d} x = - \kappa_{\beta/\alpha,\alpha, \beta} |\theta|^\beta (\omega (\theta))^{\beta/\alpha}\end{equation*}

(see (2.11) and (2.8)). This proves (4.12) and (4.11).

Proof of (2.18). Note that (4.9) also holds for $\beta > \alpha$ . Let $\{ (1-a_i)^{-1} \zeta_{\alpha, i} (\tau), \, \tau \ge 0 \}$ , $i=1,2, \dots$ , be as in (4.11). It suffices to prove that

(4.13) \begin{equation}N^{-1/\alpha} \sum_{i=1}^N (1-a_i)^{-1} \zeta_{\alpha,i} (\tau) \to_{\textnormal{fdd}} \kappa^{1/\alpha}_\alpha \zeta_\alpha (\tau).\end{equation}

Consider the one-dimensional convergence in (4.13) at $\tau = 1$ . For any $\theta \in {\mathbb{R}}$ , we have

\begin{equation*}{\mathbb{E}} \exp \bigg\{ \text{i} \theta N^{-1/\alpha}\sum_{i=1}^N (1-a_i)^{-1} \zeta_{\alpha,i}(1) \bigg\} = \bigg(1 + \frac{\Theta_N}{N} \bigg)^N,\end{equation*}

where

\begin{equation*}\Theta_N \coloneqq N{\mathbb{E}} [\!\exp \{\!-N^{-1} (1-a)^{-\alpha} |\theta|^\alpha \omega (\theta) \} - 1] \to - \kappa_\alpha|\theta|^\alpha \omega (\theta) = \log {\mathbb{E}} \text{e}^{ \text{i} \theta \kappa^{1/\alpha}_\alpha \zeta_\alpha (1)}\end{equation*}

with $\kappa_\alpha ={\mathbb{E}} (1-a)^{-\alpha} < \infty$ by the DCT. The general finite-dimensional convergence in (4.13) follows in a similar way. This proves (2.18) and completes the proof of Theorem 2.1.

Proof of Theorem 2.2. In each case of Theorem 2.2, we will prove that, for any $d \in {\mathbb{N}}$ , $0< \tau_1 < \dots < \tau_d<\infty$ , and $\theta_j \in {\mathbb{R}}$ , $j=1,\dots,d$ , as $N,n \to \infty$ ,

(4.14) \begin{equation}{\mathbb{E}} \text{e}^{ \text{i} A_{N,n}^{-1} \sum_{j=1}^d \theta_j S_{N,n} (\tau_j) } \to {\mathbb{E}} \text{e}^{\text{i} \sum_{j=1}^d \theta_j \mathcal{S}(\tau_j)},\end{equation}

where $A_{N,n} \to \infty$ denotes a sequence of normalizing constants and $\{ \mathcal{S} (\tau), \, \tau \ge 0 \}$ denotes the limit process. Since $X_1, X_2, \dots$ are independent processes, we can rewrite the left-hand side of (4.14) as $(1 + \gfrac{\Theta_{N,n}}{N} )^N$ and reduce the proof to showing that

(4.15) \begin{equation}\Theta_{N,n} \coloneqq N {\mathbb{E}} \bigg[ \exp \bigg\{ \text{i} A_{N,n}^{-1} \sum_{j=1}^d \theta_j S_{1,n} (\tau_j) \bigg\} - 1 \bigg]\to \log {\mathbb{E}} \text{e}^{ \text{i} \sum_{j=1}^{d} \theta_j \mathcal{S} (\tau_j) } =\!:\, \Theta\end{equation}

as $N,n \to \infty$ , in each case of Theorem 2.2. Conditioning on a, we have

(4.16) \begin{equation} \Theta_{N,n} = N \int_{[0,1)} \bigg[ \prod_{s \in {\mathbb{Z}}} \varphi_{{\varepsilon}} ( A_{N,n}^{-1} \vartheta_n (u,s) ) - 1 \bigg] \phi(u) \, \text{d} u \end{equation}

with

\begin{equation*}\vartheta_n (u,s) \coloneqq \sum_{j=1}^d \theta_j \sum_{t=1}^{[n\tau_j]} u^{t-s} {\textbf{1}} (s \le t),\notag\end{equation*}

where $\varphi_{\varepsilon} (\theta) \coloneqq {\mathbb{E}} \text{e}^{\text{i} \theta {\varepsilon}}$ , $\theta \in {\mathbb{R}}$ , is the characteristic function of ${\varepsilon} \in \mathcal{D}(\alpha)$ . Next, we need to split the interval $[0,1) = I_{N,n} \cup I_{N,n}^c$ with $I_{N,n} \coloneqq [0,1-u_{N,n})$ , where $u_{N,n} \to 0$ is chosen so that

\begin{equation*} \sup_{u \in I_{N,n}, s \in {\mathbb{Z}}} |\vartheta_n(u,s)| = {\mathrm{O}} (u_{N,n}^{-1})= {\mathrm{o}} (A_{N,n}) \end{equation*}

and

\begin{align*} \bigg|N {\mathbb{E}} \bigg[ \bigg(\!\exp \bigg\{ \text{i} A_{N,n}^{-1} \sum_{j=1}^d \theta_j S_{1,n} (\tau_j) \bigg\} - 1\!\bigg) {\textbf{1}} (a_1 \in I_{N,n}^c ) \bigg] \bigg| & \le C N {\mathbb{P}} ( a_1 \in I_{N,n}^c ) \\[3pt] & \le C N \int_{I_{N,n}^c} (1-u)^{\beta-1} \, \text{d} u \\[3pt] & = {\mathrm{O}} (N u_{N,n}^\beta) \\[3pt] & = {\mathrm{o}} (1) \end{align*}

is negligible. By doing so, we obtain that $h ( A_{N,n}^{-1} \vartheta_n ( u,s )) \to 1 $ uniformly in $u \in I_{N,n}, s \in {\mathbb{Z}}$ , and, taking into account (4.1), (4.15)–(4.16),

\begin{equation*}\Theta_{N,n} \sim N \int_{I_{N,n}} \bigg(\! \exp \bigg\{ - A_{N,n}^{-\alpha} \sum_{s \in {\mathbb{Z}}} | \vartheta_n (u,s) |^\alpha \omega ( \vartheta_n (u,s) ) \bigg\} - 1 \!\bigg) \phi (u) \, \text{d} u.\end{equation*}

In order to avoid this rather tedious step, from now on we will assume that $h (\theta)\equiv 1$ . That is, ${\varepsilon} =_{\textnormal{d}} \zeta_\alpha (1)$ has a stable distribution and we can take $u_{N,n}\equiv 0$ . Moreover, for simplicity of exposition, in cases (i) and (ii) of Theorem 2.2, we also assume that $\phi (u) \equiv \psi_1 (1-u)^{\beta-1}$ in (4.16). Similar simplifications are also imposed in the proof of [Reference Pilipauskaitė and Surgailis27, Theorem 2.2]. Finally, since $\omega (\theta)$ depends on the sign of $\theta $ alone, we shall assume that $\omega (\theta) \equiv 1 $ . After all, $\Theta_{N,n} $ of (4.16) reduces to

(4.17) \begin{equation}\Theta_{N,n} = \psi_1 N \int_{[0,1)} \bigg(\!\exp \bigg\{ - A_{N,n}^{-\alpha}\sum_{s \in {\mathbb{Z}}} |\vartheta_n (u,s) |^\alpha \bigg\} - 1 \!\bigg) (1-u)^{\beta -1} \, \text{d} u.\end{equation}

The only exception is the ‘short memory’ case (iii) of Theorem 2.2, where we use (4.17) with $\phi(u)$ instead of $\psi_1 (1-u)^{\beta-1}$ , $u \in [0,1)$ .

We consider each case of Theorem 2.2 separately.

Proof of Theorem 2.2 (i). Let $N,n \to \infty$ so that $\mu_{N,n} \coloneqq N^{1/\beta}/n \to \mu \in [0,\infty]$ , and set

\begin{equation*}B_{N,n} \coloneqq \begin{cases}n \\ N \\ N^{1/\beta} \\ N^{1/\beta}\end{cases}\ A_{N,n} \coloneqq n^{1+1/\alpha} \begin{cases}\mu_{N,n}^{\beta/\alpha} & \text{if } \mu = \infty, \ 1 < \beta < \alpha,\\(\mu_{N,n} \log \mu_{N,n})^{1/\alpha} & \text{if } \mu = \infty, \ 1 = \beta < \alpha,\\\mu_{N,n}^{1/\alpha} & \text{if } \mu = \infty, \ 0 < \beta < \min\!(\alpha,1),\\\mu_{N,n} & \text{if } \mu \in [0, \infty), \ 0 < \beta < \alpha.\end{cases}\end{equation*}

( $B_{N,n}$ and $A_{N,n}$ are simultaneously defined in the above four cases of $\mu, \alpha, \beta $ .) Note that $A_{N,n}$ agree with respective normalizations in Theorem 2.2(i). After the change of variable $B_{N,n} (1-u) = x$ , (4.17) can be rewritten as

\begin{equation*}\Theta_{N,n} = (\psi_1 N/B^\beta_{N,n}) \int_0^{B_{N,n}} (\text{e}^{ - K_{N,n}(x)} -1 ) x^{\beta -1} \, \text{d} x ,\end{equation*}

where

(4.18) \begin{align}K_{N,n}(x) &= A_{N,n}^{-\alpha} \sum_{s \le 0} \bigg| \vartheta_n \bigg( 1-\frac{x}{B_{N,n}}, s \bigg) \bigg|^\alpha+ A_{N,n}^{-\alpha} \sum_{s > 0} \bigg| \vartheta_n \bigg( 1-\frac{x}{B_{N,n}}, s \bigg) \bigg|^\alpha\notag\\[3pt] &=\!:\, K_{N,n}^- (x) + K_{N,n}^+ (x).\end{align}

Proof of (2.23) (case $\mu \in (0,\infty)$ , $0 < \beta < \alpha$ ). The right-hand side of (4.15) can be written as the integral

\begin{equation*}\Theta =\psi_1 \int_0^\infty ( \text{e}^{-K_\mu (x)} - 1 ) x^{\beta-1} \, \text{d} x\end{equation*}

(see (2.13)), where

(4.19) \begin{equation}K_\mu (x) \coloneqq \mu \int_{{\mathbb{R}}} \bigg| \sum_{j=1}^d \theta_j f_{\tau_j/\mu} (x,s) \bigg|^\alpha\, \text{d} s\end{equation}

and $f_\tau (x,s)$ as in (2.5). Using $B_{N,n} = N^{1/\beta}$ and writing the sums over integers s, t in (4.18), (4.16) as integrals, after the change of variables $s \to N^{1/\beta}s$ , $t \to N^{1/\beta} t$ , the left-hand side of (4.15) becomes

\begin{equation*}\Theta_{N,n} = \psi_1 \int_0^{N^{1/\beta}} (\text{e}^{ - K_{N,n} (x) } - 1) x^{\beta-1} \, \text{d} x\end{equation*}

with

\begin{equation*}K_{N,n} (x)= \mu_{N,n} \int_{\mathbb{R}} | \tilde \vartheta_{N,n} (x,s) |^\alpha\, \text{d} s,\end{equation*}

where

(4.20) \begin{align}\tilde \vartheta_{N,n} (x,s)&\coloneqq \frac{1}{N^{1/\beta}} \vartheta_n \bigg( 1 - \frac{x}{N^{1/\beta}}, [N^{1/\beta} s] \bigg)\notag \\[3pt] &= \sum_{j=1}^d \theta_j\int_{{\mathbb{R}}}\bigg( 1 - \frac{x}{N^{1/\beta}} \bigg)^{[N^{1/\beta} t] - [N^{1/\beta} s]} \, {\textbf{1}} ( \max\!([N^{1/\beta} s],1) \le [N^{1/\beta} t] \le [n \tau_j] ) \, \text{d} t\notag \\[3pt] &\to \sum_{j=1}^d \theta_j f_{\tau_j/\mu} (x,s)\end{align}

for any $x \in {\mathbb{R}}_+$ , $s \in {\mathbb{R}}$ . Using $1-z \le \text{e}^{-z}$ , $z \in [0,1]$ , we get the dominating bound $| \tilde \vartheta_{N,n} (x,s) | \le C f_{2 \tau_d/\mu} (x,s)$ , $x \in (0, N^{1/\beta}]$ , $s \in {\mathbb{R}}$ , implying $K_{N,n} (x) \to K_\mu (x)$ , $x \in {\mathbb{R}}_+$ , by the DCT. Using (4.3), we can extend the last dominating bound to

\begin{equation*}|K_{N,n}(x)| \le C \int_{\mathbb{R}} |f_{2\tau_d/\mu}(x,s)|^\alpha \, \text{d} s,\quad x \in (0, N^{1/\beta}],\end{equation*}

where the last integral is estimated in (4.29). Then, another application of the DCT yields the convergence in (4.15), proving (2.23).

Proof of (2.19) (case $\mu = \infty$ , $1 < \beta < \alpha$ ). By (2.6), the right-hand side of (4.15) is equal to

\begin{equation*}- \psi_1 \int_0^\infty K_1 (x) x^{\beta-1} \, \text{d} x,\end{equation*}

where $K_1 (x)$ is as in (4.19) (with $\mu = 1$ ). After a change of variable, the left-hand side of (4.15) can be written as

\begin{equation*}\Theta_{N,n} = \psi_1 \int_0^n \mu_{N,n}^\beta ( \text{e}^{ - \mu_{N,n}^{-\beta} K_{N,n} (x) } - 1 ) x^{\beta-1} \, \text{d} x,\end{equation*}

with

\begin{equation*}K_{N,n} (x) \coloneqq \int_{{\mathbb{R}}} | \tilde \vartheta_{N,n} (x,s) |^\alpha \, \text{d} s\end{equation*}

and

\begin{equation*}\tilde \vartheta_{N,n} (x,s) \coloneqq \frac{1}{n} \vartheta_n \bigg( 1 - \frac{x}{n}, [n s] \bigg) \to \sum_{j=1}^d \theta_j f_{\tau_j} (x,s)\end{equation*}

for any $x \in {\mathbb{R}}_+$ , $s \in {\mathbb{R}}$ (to justify the last relationship, use (4.20) with $N^{1/\beta}$ replaced by n). Therefore,

\begin{equation*}K_{N,n}(x) \to K_1 (x),\ \mu_{N,n}^\beta ( \text{e}^{ - \mu_{N,n}^{-\beta} K_{N,n} (x) } - 1 ) \to - K_1 (x), \quad x \in {\mathbb{R}}_+,\end{equation*}

and, finally, the convergence in (4.15) follows using the DCT similarly to the proof before. This proves (2.19).

Proof of (2.20) (case $\mu = \infty$ , $0 < \beta < \min\!(\alpha,1)).$ The right-hand side of (4.15) can be written as

\begin{equation*}\Theta =-\kappa_{\beta,\alpha,\beta} \bigg| \sum_{j=1}^d \theta_j \tau_j \bigg|^{\alpha \beta}= \psi_1 \int_0^\infty \bigg( \exp \bigg\{\!- \frac{1}{\alpha x} \bigg|\sum_{j=1}^d \theta_j \tau_j \bigg|^\alpha\bigg\} - 1 \bigg) x^{\beta-1} \, \text{d} x\end{equation*}

(see (2.8)–(2.11)). On the other hand, for the left-hand side of (4.15), after a change of variable, we get

\begin{equation*}\Theta_{N,n} = \psi_1 \int_0^{N^{1/\beta}} ( \text{e}^{-K_{N,n}(x)} - 1 ) x^{\beta-1} \, \text{d} x,\end{equation*}

where $K_{N,n} (x) = K_{N,n}^- (x) + K_{N,n}^+ (x)$ is given in (4.18) with $B_{N,n} = N^{1/\beta}$ . Rewrite

\begin{equation*}K_{N,n}^- (x) = \int_{{\mathbb{R}}} | \tilde \vartheta_{N,n} (x,s) |^\alpha\, \text{d} s,\end{equation*}

where

\begin{align*}\tilde \vartheta_{N,n} (x,s)&\coloneqq \sum_{j=1}^d \theta_j \int_{{\mathbb{R}}} \bigg( 1-\frac{x}{N^{1/\beta}} \bigg)^{[nt]-[N^{1/\beta}s]} {\textbf{1}} ( 1 \le [nt] \le [n\tau_j], \, [N^{1/\beta} s] \le 0 ) \, \text{d} t\\*&\to \sum_{j=1}^d \theta_j \tau_j \text{e}^{x s} {\textbf{1}} (s \le 0)\end{align*}

in view of $n/N^{1/\beta} \to 0$ , for any $x > 0$ , $s \in {\mathbb{R}}, s \ne 0$ . We have the dominating bound $|\tilde \vartheta_{N,n}(x,s)| \le C \text{e}^{xs} {\textbf{1}} (s \le C)$ , $x \in (0, N^{1/\beta}]$ , $s \in {\mathbb{R}}$ , whence

(4.21) \begin{equation}K_{N,n}^- (x) \to \frac{1}{\alpha x} \bigg| \sum_{j=1}^d \theta_j \tau_j \bigg|^\alpha \quad \text{and} \quad |K_{N,n}^- (x)| \le \frac{C}{x}\end{equation}

for $x \in (0,N^{1/\beta}]$ . Relation (4.2) implies

\begin{equation*}|K_{N,n}^+ (x)| \le \frac{C n}{A_{N,n}^{\alpha}} \min \bigg( n, \frac{N^{1/\beta}}{x} \bigg)^{\alpha} \le \frac{C}{\mu_{N,n}} \min \bigg(1, \frac{\mu_{N,n}}{x} \bigg)^{\min\!({\alpha},1)} = {\mathrm{o}} (1)\end{equation*}

uniformly in $x > 0$ as $N,n\to\infty$ . Moreover, we conclude that $|K_{N,n}(x)| \le C x^{-\min\!(\alpha,1)}$ for $x \in (1,N^{1/\beta}]$ . Since $0 < \beta < \min\!(\alpha,1)$ , in view of (4.3), the DCT implies (4.15), proving (2.20).

Proof of (2.21) (case $\mu = \infty$ , $1=\beta < \alpha$ ). Write the right-hand side and left-hand side of (4.15) respectively as

\begin{equation*}\Theta = - \frac{\psi_1}{\alpha} \bigg| \sum_{j=1}^d \theta_j \tau_j \bigg|^\alpha\end{equation*}

and

\begin{equation*}\Theta_{N,n} = N \psi_1\int_{[0,1)} \bigg( \exp \bigg\{ - \frac{ K_{N,n}(u)}{(1-u) N \log \pgfrac{N}{n}} \bigg\} - 1 \bigg) \, \text{d} u \end{equation*}

with

\begin{equation*} K_{N,n}(u) \coloneqq n^{-\alpha} (1-u) \sum_{s \in {\mathbb{Z}}} |\vartheta_n (u,s)|^\alpha.\end{equation*}

For given $\delta \in (0,1)$ , split

\begin{align*}\Theta_{N,n} &= N \psi_1 \bigg\{\int_0^{1- \gfrac{\delta}{n}} + \int_{1- \gfrac{\delta}{N}}^1 + \int_{1-\gfrac{\delta}{n}}^{1- \gfrac{\delta}{N}}\bigg\}\bigg( \exp \bigg\{ - \frac{K_{N,n}(u)}{(1-u) N \log \pgfrac{N}{n}} \bigg\} - 1 \bigg) \, \text{d} u\\[3pt] &=\!:\,\sum_{i=1}^3 \Theta_{N,n,\delta}^i.\end{align*}

By (4.2) and (4.3), for every $\delta>0$ ,

\begin{align*}|\Theta_{N,n,\delta}^1| &\le \frac{C}{n^\alpha \log \pgfrac{N}{n}} \int_{0}^{1-\gfrac{\delta}{n}} \bigg( \frac{1}{1-u} \min \bigg(n, \frac{1}{1-u} \bigg)^\alpha+ n \min \bigg(n,\frac{1}{1-u}\bigg)^\alpha \bigg) \, \text{d} u\\[4pt] &\le \frac{C}{n^\alpha \log \pgfrac{N}{n}} \int_{\delta}^{n} \bigg( \frac{1}{x} \min \bigg( n,\frac{n}{x} \bigg)^\alpha + \min \bigg(n,\frac{n}{x} \bigg)^\alpha \bigg) \, \text{d} x \\[4pt] & = {\mathrm{o}} (1)\end{align*}

as $N/n \to \infty $ . Also, $|\Theta_{N,n,\delta}^2| \le C\delta $ can be made arbitrarily small by a suitable choice of $\delta >0$ . Consider

\begin{align*}\Theta_{N,n,\delta}^3 &= \psi_1 \int_{\delta}^{\delta \pgfrac{N}{n}} \bigg( \exp \bigg\{ - \frac{K_{N,n} (1- \gfrac{x}{N} )}{x \log \pgfrac{N}{n}} \bigg\} - 1 \bigg) \, \text{d} x\\[4pt] &\sim - \psi_1 \int_{\delta}^{\delta \pgfrac{N}{n}} \frac{K_{N,n} (1- \gfrac{x}{N} )}{x \log \pgfrac{N}{n}} \, \text{d} x \\[4pt] & =\!:\, \Theta_{N,n,\delta}^4 {,}\end{align*}

since

\begin{equation*}|\Theta_{N,n,\delta}^3 - \Theta_{N,n,\delta}^4| \le C \int_\delta^{\delta (N/n)} \frac{\text{d} x}{(x \log (N/n))^2}= {\mathrm{o}} (1),\quad \text{as $N/n \to \infty$},\end{equation*}

follows from (4.3) and the bound

\begin{equation*}\sup_{0< x < \delta N/n} \bigg| K_{N,n}\bigg(1-\frac{x}{N}\bigg)\bigg| \le C,\end{equation*}

which is a consequence of (4.2). Rewrite $\Theta $ similarly to $\Theta_{N,n,\delta}^4$ , namely,

\begin{equation*}\Theta = - \psi_1 K \int_{\delta}^{\delta \pgfrac{N}{n}} \frac{\text{d} x}{x \log \pgfrac{N}{n}} \quad \text{with }\ K \coloneqq \frac{1}{\alpha} \bigg| \sum_{j=1}^d \theta_j \tau_j \bigg|^\alpha.\end{equation*}

Thus, (2.21) follows (cf. (4.7)) from

(4.22) \begin{equation}\limsup_{N,n, N/n \to \infty} \sup_{\delta < x < \delta (N/n)} \bigg|K_{N,n} \bigg( 1-\frac{x}{N} \bigg)- K\bigg| \le \epsilon(\delta) {,}\end{equation}

where $\lim_{\delta \to 0} \epsilon(\delta) = 0$ . To this end, write

\begin{equation*}K_{N,n} \bigg(1- \frac{x}{N} \bigg) = \tilde K\bigg( \frac{N}{x}, n\bigg)\end{equation*}

as in (4.7) above, where

\begin{equation*}\tilde K(y,z)\coloneqq \int_{{\mathbb{R}}} \bigg| \sum_{j=1}^d \theta_j \int_{{\mathbb{R}}} \bigg( 1- \frac{1}{y} \bigg)^{[z t]-[ys]} {\textbf{1}}(1\vee [ys] \le [zt] \le [z\tau_j]) \, \text{d} t \bigg|^\alpha \, \text{d} s,\quad y, z >0.\end{equation*}

Then, using the DCT similarly as in (4.8) above, it follows that $\lim_{y,z,y/z \to \infty} \tilde K(y,z) = K $ , implying that $\limsup_{y,z,y/z \to \infty} \sup_{y/z > 1/\delta} |\tilde K(y,z) - K| \le \epsilon (\delta)$ with $\epsilon(\delta) $ as in (4.22), hence (4.22), thus completing the proof of (2.21).

Proof of (2.22) (case $\mu = 0$ , $0< \beta < \alpha$ ). The right-hand side of (4.15) can be written as

\begin{align*}\Theta&=\log {\mathbb{E}} \text{e}^{ \text{i} W^{1/\alpha}_{\beta/\alpha, \alpha,\beta} \sum_{i=1}^d (\!\sum_{j=i}^d \theta_j ) ( \zeta_\alpha (\tau_i) - \zeta_\alpha (\tau_{i-1}) ) }\notag \\[3pt] &= \log {\mathbb{E}} \text{e}^{ - W_{\beta/\alpha,\alpha,\beta}\sum_{i=1}^d (\tau_i - \tau_{i-1}) |\sum_{j=i}^d \theta_j |^\alpha} \\[3pt] & = - \kappa_{\beta/\alpha,\alpha,\beta} \bigg( \sum_{i=1}^d (\tau_i - \tau_{i-1} ) \bigg| \sum_{j=i}^d \theta_j \bigg|^\alpha\bigg)^{\beta/\alpha}\notag \\[3pt] &= \psi_1 \int_0^\infty \bigg( \exp \bigg\{ - \frac{1}{x^\alpha} \sum_{i=1}^d (\tau_i-\tau_{i-1}) \bigg| \sum_{j=i}^d \theta_j \bigg|^\alpha\bigg\} - 1 \bigg) x^{\beta-1} \, \text{d} x\end{align*}

with $\tau_0 \coloneqq 0$ and $\kappa_{\beta/\alpha,\alpha,\beta}$ as in (2.11). After a change of variable, we get

\begin{equation*}\Theta_{N,n} = \psi_1 \int_0^{N^{1/\beta}} ( \text{e}^{-K_{N,n} (x)} - 1 ) x^{\beta-1} \, \text{d} x,\end{equation*}

where $K_{N,n} (x) = K_{N,n}^- (x)+K_{N,n}^+ (x)$ is given as in (4.18) with $B_{N,n} = N^{1/\beta}$ . Rewrite

\begin{equation*}K_{N,n}^+ (x) = \int_{{\mathbb{R}}} | \tilde \vartheta_{N,n} (x,s) |^\alpha\, \text{d} s,\end{equation*}

where

\begin{align*}\tilde \vartheta_{N,n} (x,s)&\coloneqq \frac{1}{x} \sum_{j=1}^d \theta_j \bigg(1- \bigg( 1-\frac{x}{N^{1/\beta}} \bigg)^{[n\tau_j]-[ns]+1} \bigg) {\textbf{1}} (0 < [ns] \le [n\tau_j] )\\[3pt] &\to \frac{1}{x} \sum_{j=1}^d \theta_j {\textbf{1}} (0 < s \le \tau_j)\end{align*}

for $x \in {\mathbb{R}}_+$ , $s \in {\mathbb{R}}$ , since $n/N^{1/\beta} \to \infty$ . Therefore,

\begin{equation*}K_{N,n}^+ (x) \to \frac{1}{x^\alpha} \sum_{i=1}^d (\tau_i-\tau_{i-1}) \bigg| \sum_{j=i}^d \theta_j \bigg|^\alpha \quad\text{for $x \in {\mathbb{R}}_+$.}\end{equation*}

By (4.2), it follows that

\begin{equation*}| K_{N,n}^- (x) | \le C A_{N,n}^{-\alpha} \bigg( \frac{x}{N^{1/\beta}} \bigg)^{-(1+\alpha)} = C \mu_{N,n} x^{-(1+\alpha)} = {\mathrm{o}} (1)\quad\text{for $x \in {\mathbb{R}}_+$.}\end{equation*}

Moreover, $|K_{N,n}(x)| \le C x^{-\alpha}, \ x \in (1,N^{1/\beta}]$ . Since $0 < \beta < \alpha$ , the DCT and (4.3) imply $\Theta_{N,n} \to \Theta$ , proving (2.22), thereby completing the proof of Theorem 2.2(i).

Proof of Theorem 2.2(iii). We use the representation in (4.17) with $A_{N,n} = (N n)^{1/\alpha} $ and $\phi(u)$ instead of $\psi_1 (1-u)^{\beta-1}$ , $u \in [0,1)$ . We have

\begin{equation*} \Theta_{N,n} = \int_0^1 N (\text{e}^{-N^{-1} K_{n}(u)} - 1) \phi(u) \, \text{d} u, \end{equation*}

where

\begin{equation*} K_{n} (u) = K^-_n(u) + K^+_n(u) \coloneqq n^{-1} \sum_{s \le 0} | \vartheta_n (u,s) |^\alpha + n^{-1} \sum_{s > 0} | \vartheta_n (u,s) |^\alpha{,} \end{equation*}

with $\vartheta_n (u,s)$ defined in (4.16). Rewrite

\begin{equation*}K_{n}^+ (u) = \int_{{\mathbb{R}}} |\vartheta_n(u,[ns])|^\alpha\, \text{d} s,\end{equation*}

where

\begin{align*} \vartheta_n (u,[ns]) & = (1-u)^{-1} \sum_{j=1}^d \theta_j (1-u^{[n\tau_j]-[n s] +1}) {\textbf{1}} (0 < [n s] \le [n \tau_j])\\[4pt] & \to (1-u)^{-1} \sum_{j=1}^d \theta_j {\textbf{1}} (0 < s \le \tau_j)\quad\text{for $u \in [0,1),\ s \in {\mathbb{R}}$.} \end{align*}

Hence

(4.23) \begin{equation}K_{n}^+ (u) \to (1-u)^{-\alpha} \sum_{i=1}^d (\tau_i - \tau_{i-1}) \bigg| \sum_{j=i}^{d} \theta_j \bigg|^\alpha\quad \text{for $u \in [0,1)$,}\end{equation}

where $\tau_0 \coloneqq 0$ . From (4.2), for $u \in [0,1)$ , we further obtain that $K^-_n(u)\to 0$ and that

(4.24) \begin{equation} |K_{n}^- (u)| \le\frac{C}{n(1-u)} \min \bigg(n, \frac{1}{1-u} \bigg)^\alpha\le C \begin{cases} (1-u)^{-\alpha} &\alpha \ge 1, \\(1-u)^{-1} &\alpha < 1,\end{cases}\end{equation}

and $|K_{n}^+(u)| \le C (1-u)^{-\alpha}$ , hence $|K_{n} (u)| \le C(1-u)^{-\max\!(\alpha,1)} =\!:\, \bar K(u)$ , where

\begin{equation*}\int_0^1 \bar K(u) \phi (u) \, \text{d} u < \infty\end{equation*}

due to the fact that $\beta > \max (\alpha, 1)$ . Then, by the DCT, we conclude that

\begin{equation*}\Theta_{N,n} \to - \kappa_\alpha \sum_{i=1}^d (\tau_i - \tau_{i-1}) \bigg| \sum_{j=i}^{d} \theta_j \bigg|^\alpha= \log {\mathbb{E}} \text{e}^{\text{i} \kappa_\alpha^{1/\alpha} \sum_{j=1}^{d} \theta_j \zeta_\alpha (\tau_j) },\end{equation*}

where (recall) $\kappa_\alpha = {\mathbb{E}} (1-a)^{-\alpha} < \infty$ . Theorem 2.2(iii) is proved.

Proof of Theorem 2.2 (ii). Recall that $0< \alpha < \beta < 1 $ . Set $\mu_{N,n} \coloneqq N^{1/(\gamma\beta)}/n \to \mu \in [0,\infty]$ , $A_{N,n} \coloneqq N^{1/(\alpha \beta)} n$ if $\mu = \infty$ ; and $A_{N,n} \coloneqq (Nn)^{1/\alpha}$ if $\mu \in [0, \infty)$ . Consider separately terms $S_{N,n}^\pm (\tau)$ in the decomposition $S_{N,n}(\tau) = S_{N,n}^- (\tau) + S_{N,n}^+ (\tau)$ , where

\begin{equation*}S_{N,n}^- (\tau) \coloneqq \sum_{i=1}^N \bigg( \sum_{t=1}^{[n\tau]} a_i^t \bigg) X_i(0), \quad S_{N,n}^+ (\tau) \coloneqq \sum_{i=1}^N \sum_{s>0} \bigg( \sum_{t=1}^{[n\tau]} a_i^{t-s} {\textbf{1}} (s \le t) \bigg) {\varepsilon}_i (s).\end{equation*}

From the proofs of (2.20), (2.27) (in particular, (4.21), (4.23), (4.24)) we see that

(4.25) \begin{alignat}{3}N^{-1/(\alpha \beta)} n^{-1} S^-_{N,n} (\tau) &\to_{\textnormal{fdd}} V_{\alpha,\beta}\, \tau & \quad & \text{as } N,n, N^{1/\beta}/n \to \infty,\ \text{for } 0 < \beta < 1, \end{alignat}

(4.26) \begin{alignat}{3}(N n)^{-1/\alpha} S^+_{N,n} (\tau) &\to_{\textnormal{fdd}} \kappa^{1/\alpha}_\alpha \zeta_\alpha (\tau) & \quad & \text{as } N,n \to \infty\ \text{in {an} arbitrary way, for } \beta > \alpha.\end{alignat}

Proof of (2.24) (case $\mu = \infty$ ). This follows from (4.25) and (4.26) since $\mu_{N,n} \to \infty $ implies $(Nn)^{1/\alpha} = {\mathrm{o}} (A_{N,n})$ .

Proof of (2.25) (case $\mu = 0$ ). This follows in view of (4.26), if we prove that

(4.27) \begin{equation}S_{N,n}^- (\tau) = o_{\textnormal{p}} ( A_{N,n} ).\end{equation}

For any $\theta \in {\mathbb{R}}$ , consider

\begin{align*}\Theta_{N,n} &\coloneqq N {\mathbb{E}} [\!\exp \{ \text{i} \theta A_{N,n}^{-1} S_{1,n}^- (\tau) \} - 1 ]\\[2pt] &= \psi_1 N \int_0^1 \bigg( \exp \bigg\{ - A_{N,n}^{-\alpha} \sum_{s \le 0} | c_{[n\tau]} (u,s) |^\alpha |\theta|^\alpha \bigg\} - 1 \bigg) (1-u)^{\beta-1} \, \text{d} u,\end{align*}

where $c_{[n\tau]}(u,s)$ is given in (4.2). Using (4.2)–(4.3) and changing a variable, we obtain

\begin{align*}|\Theta_{N,n}| &\le C N \int_0^1 \min \bigg( 1, \frac{n^{\alpha-1}}{N(1-u)} \bigg) (1-u)^{\beta-1} \, \text{d} u\\[3pt] &\le C \int_0^{N^{1/\beta}} \min \bigg( 1, \frac{\mu_{N,n}^{1-\alpha}}{x} \bigg) x^{\beta-1} \, \text{d} x \\[3pt] & = {\mathrm{o}} (1).\end{align*}

This proves (4.27), and hence (2.25).

Proof of (2.26) (case $\mu \in (0,\infty)$ ). For N, n large enough, decompose $\Theta_{N,n} = \Theta_{N,n}^- + \Theta_{N,n}^+$ in (4.17), with $\psi_1 (1-u)^{\beta -1}$ replaced by $\phi(u)$ , as

\begin{align*}\Theta_{N,n}^- &\coloneqq N \int_0^1 ( \text{e}^{ - K_{N,n}^- (u) } - 1 ) \phi (u) \, \text{d} u,\\[3pt] \Theta_{N,n}^+ &\coloneqq N \int_0^1 \text{e}^{ - K_{N,n}^- (u) } ( \text{e}^{ - K_{N,n}^+ (u) } - 1 ) \phi (u) \, \text{d} u,\end{align*}

where

\begin{equation*}K_{N,n}^- (u) \coloneqq A_{N,n}^{-\alpha} \sum_{s\le 0} |\vartheta_n(u,s)|^\alpha,\quad K_{N,n}^+ (u) \coloneqq A_{N,n}^{-\alpha} \sum_{s > 0} |\vartheta_n(u,s)|^\alpha.\end{equation*}

Since $\mu_{N,n} \to \mu \in (0,\infty)$ implies $N^{1/\beta}/n \to \infty$ , by (4.25) we have

\begin{equation*}\Theta_{N,n}^- \to \log {\mathbb{E}} \text{e}^{\text{i} \mu^{(1/\alpha)- 1} (\!\sum_{j=1}^d \theta_j \tau_j) V_{\alpha,\beta}}.\end{equation*}

Next, using $K_{N,n}^- (u) \to 0$ , $|\text{e}^{-K_{N,n}^- (u)} | \le C$ , $u \in [0,1)$ , similarly to the proof of (4.26), we obtain

\begin{equation*} \Theta_{N,n}^+ \to \log {\mathbb{E}} \text{e}^{\text{i} \kappa^{1/\alpha}_\alpha \sum_{j=1}^d \theta_j \zeta_\alpha (\tau_j) }. \end{equation*}

Hence, (2.26) follows, including the independence of $V_{\alpha,\beta}$ and $\{\zeta_\alpha(\tau),\, \tau \ge 0\}$ . Theorem 2.2 is proved.

Proof of Proposition 3.1. As noted in Section 3, for $\alpha =2 $ , the proposition is proved in [Reference Pilipauskaitė and Surgailis27, Propositions 3.1, 3.2]. The subsequent proof for $0 < \alpha < 2$ uses a similar argument.

For (i), write $\mathcal{Z}_{\alpha,\beta} (\tau) = \mathcal{Z}^-_{\alpha,\beta} (\tau) + \mathcal{Z}^+_{\alpha,\beta}(\tau)$ with

\begin{align*} \mathcal{Z}^+_{\alpha,\beta} (\tau) & \coloneqq \int_{[1,\infty) \times D({\mathbb{R}})} z (\tau;\ x ) (N(\text{d} x, \text{d} \zeta)- \nu (\text{d} x, \text{d} \zeta) {\textbf{1}} (\alpha > 1) ), \\[3pt] \mathcal{Z}^-_{\alpha,\beta} (\tau)& \coloneqq \int_{(0,1) \times D({\mathbb{R}})} z ({\tau;}\ x ) N(\text{d} x, \text{d} \zeta).\end{align*}

Next, let $I(p,\tau) \coloneqq I^-(p,\tau) + I^+(p,\tau), $ where

\begin{equation*}I^+(p,\tau) \coloneqq \int_1^\infty {\mathbb{E}}_\alpha |z(\tau;x)|^p x^{\beta-1} \, \text{d} x,\quad I^-(p,\tau) \coloneqq \int_0^1 {\mathbb{E}}_\alpha |z(\tau; \, x)|^p x^{\beta-1} \, \text{d} x.\end{equation*}

Then $\mathcal{Z}^\pm_{\alpha,\beta} (\tau)$ are well-defined if $I^\pm (p,\tau) < \infty $ for some $0< p < \alpha$ , in which case

(4.28) \begin{equation}{\mathbb{E}} |\mathcal{Z}^\pm_{\alpha,\beta}(\tau)|^p \le C I^\pm (p, \tau)\end{equation}

with $C = C(p)$ depending on p alone; see [27, (3.3)]. By a well-known property of an $\alpha$ -stable stochastic integral in (3.1),

\begin{equation*} {\mathbb{E}}_\alpha | z(\tau;x) |^p = C \bigg(\int_{\mathbb{R}} |f_\tau (x,s)|^\alpha \, \text{d} s\bigg)^{p/\alpha}\quad\text{{for all} } p \in(0, \alpha)\end{equation*}

(see [Reference Samorodnitsky and Taqqu34, Property 1.2.17, Proposition 3.4.1]). Then, using (4.3), we obtain

(4.29) \begin{align}\int_{{\mathbb{R}}} |f_\tau (x,s)|^\alpha \, \text{d} s&= \frac{(1-\text{e}^{-x\tau})^\alpha}{\alpha x^{1+\alpha}} + \frac{1}{x^{\alpha}} \int_0^\tau ( 1-\text{e}^{-xs} )^\alpha \, \text{d} s\notag\\[3pt] &\le \frac{1}{\alpha} \min \bigg( \frac{\tau^\alpha}{x}, \frac{1}{x^{1+\alpha}} \bigg) + \min \bigg( \frac{\tau}{x^\alpha}, \tau^{1+\alpha} \bigg).\end{align}

First let $1< \alpha < 2$ . Then (4.29) simplifies to

\begin{equation*}\int_{\mathbb{R}} |f_\tau (x,s)|^\alpha \, \text{d} s \le C \min \bigg( \frac{\tau^\alpha}{x}, \frac{\tau}{x^\alpha} \bigg) {,}\end{equation*}

implying

(4.30) \begin{equation}I(p,\tau)\le C \bigg(\tau^p \int_0^{1/\tau} x^{\beta-1-p/\alpha} \, \text{d} x + \tau^{p/\alpha} \int_{1/\tau}^\infty x^{\beta-1 -p} \, \text{d} x \bigg)\le C \tau^{p + p/\alpha - \beta}\end{equation}

for $\beta < p < \min (\alpha, \alpha \beta) = \alpha \min\!(1, \beta),$ with $C >0$ independent of $\tau >0$ . Obviously, for given $0< \beta < \alpha $ , such a p exists, implying the existence of the Poisson stochastic integrals $\mathcal{Z}^\pm_{\alpha,\beta} (\tau)$ and $\mathcal{Z}_{\alpha,\beta} (\tau)$ .

Next, let $0< \alpha \le 1 $ . Here, we need to discuss the existence of $\mathcal{Z}^\pm_{\alpha,\beta} (\tau)$ separately. From (4.29), we have

(4.31) \begin{equation} \begin{aligned}I^+(p^+,1) & \le C \int_1^\infty x^{\beta-1-p^+} \, \text{d} x \le \infty, & & \beta < p^+\! < \alpha, \\[3pt]\ I^-(p^-,1) & \le C \int_0^1 x^{\beta-1-p^-/\alpha} \, \text{d} x\le \infty, & & p^- < \alpha\beta < \alpha.\end{aligned}\end{equation}

Clearly, for any $0< \beta < \alpha \le 1, $ such $p^\pm $ satisfying (4.31) exist, implying the existence of $\mathcal{Z}^\pm_{\alpha,\beta} (\tau)$ and $\mathcal{Z}_{\alpha,\beta} (\tau)$ for $\tau=1$ , and the last result extends to all $\tau >0$ in an obvious way. The stationarity of increments is immediate from (2.13). Infinite divisibility and the form of the characteristic function in (2.13) follow from general properties of Poisson stochastic integrals; see e.g. [Reference Pilipauskaitė and Surgailis27, (3.1)].

Part (ii) follows from (4.30), (4.31), and (4.28).

Part (iii) follows from (ii) and ${\mathbb{E}} |\mathcal{Z}_{\alpha,\beta}(\tau)| \le CI(1,\tau) < \infty $ since $1< \alpha \min (\beta, 1)$ is equivalent to $\alpha > 1, \alpha \beta > 1 $ .

Part (iv) follows from Kolmogorov’s criterion, stationarity of increments of $\mathcal{Z}_{\alpha,\beta}$ , and (4.28) and (4.30) since, for $1/\alpha < \beta < \alpha$ , we can find p sufficiently close to $\alpha \min\!(\beta, 1)$ such that the exponent of $\tau$ on the right-hand side of (4.30) is greater than 1: $p+p/\alpha -\beta > 1$ .

For (v), for brevity we assume $\omega (\theta) \equiv 1$ , $\psi_1 = 1$ and restrict the proof to one-dimensional convergence at $\tau > 0$ . Let $1 < \beta < \alpha$ and $H \coloneqq 1-\gpfrac{\beta-1}{\alpha} > 0$ . For any $\theta \in {\mathbb{R}}$ , we have

\begin{equation*}{\mathbb{E}} \text{e}^{ \text{i} \theta c^{-H} \mathcal{Z}_{\alpha,\beta} (c \tau) }= \exp \bigg\{ \int_{{\mathbb{R}}_+} \bigg(\!\exp \bigg\{ - c^{\beta} | \theta|^\alpha \int_{{\mathbb{R}}} |f_{\tau} (c x,s)|^\alpha \, \text{d} s \bigg\} -1 \!\bigg) x^{\beta-1} \, \text{d} x \bigg\}{,}\end{equation*}

since

\begin{equation*}\int_{\mathbb{R}} |f_{c \tau }(x,s)|^\alpha \, \text{d} s = c^{1+\alpha} \int_{\mathbb{R}} |f_{\tau }(c x,s)|^\alpha \, \text{d} s.\end{equation*}

By change of variables, we further rewrite

\begin{align*}{\mathbb{E}} \text{e}^{ \text{i} \theta c^{-H} \mathcal{Z}_{\alpha,\beta}(c \tau) }&= \exp \bigg\{ \int_{{\mathbb{R}}_+} c^{-\beta} \bigg( \exp \bigg\{ - c^{\beta} | \theta|^\alpha \int_{{\mathbb{R}}} |f_{\tau} (x,s)|^\alpha \, \text{d} s \bigg\} -1 \bigg) x^{\beta-1} \, \text{d} x \bigg\}\\[3pt] &\to \exp \bigg\{ - | \theta|^\alpha \int_{{\mathbb{R}}_+ \times {\mathbb{R}}} |f_{\tau} (x,s)|^\alpha x^{\beta-1} \, \text{d} x \, \text{d} s \bigg\} \\[3pt] & ={\mathbb{E}} \text{e}^{\text{i} \theta \Lambda_{\alpha,\beta} (\tau)}, \quad c \to 0,\end{align*}

where the convergence follows by the DCT (the domination can be verified using (4.3) and (4.29)).

Next, let $0 < \beta < \min\!(\alpha, 1)$ . For any $\theta \in {\mathbb{R}}$ , we have

\begin{equation*}{\mathbb{E}} \text{e}^{\text{i} \theta c^{-1} \mathcal{Z}_{\alpha,\beta}(c \tau)} = \exp \bigg\{ \int_{{\mathbb{R}}_+} ( \text{e}^{ - |\theta|^\alpha K_c (x) } - 1 ) x^{\beta-1} \, \text{d} x \bigg\},\end{equation*}

where

\begin{align*}K_c (x) &\coloneqq \frac{1}{c^\alpha} \int_{{\mathbb{R}}} |f_{c\tau} (x,s)|^\alpha \, \text{d} s\\[3pt] &= \frac{1}{\alpha x} \bigg( \frac{1-\text{e}^{-c x \tau}}{cx} \bigg)^\alpha + \int_0^{c \tau} \bigg( \frac{1 - \text{e}^{-xs}}{cx} \bigg)^\alpha \, \text{d} s \\[3pt] & \to \frac{\tau^\alpha}{\alpha x}, \quad c \to 0.\end{align*}

For $K_c(x)$ , we can find a dominating function using (4.3) and (4.29), because the latter inequality gives $K_c (x) \le C x^{-1}$ if $\alpha \ge 1$ and $K_c (x) \le C \max (x^{-1}, c^{1-\alpha} x^{-\alpha}) \le C x^{-\alpha}$ if $\alpha < 1$ for $x > 1$ . Then, by the DCT, we obtain

\begin{equation*} {\mathbb{E}} \text{e}^{\text{i} c^{-1} \mathcal{Z}_{\alpha,\beta}(c \tau)} \to \exp \bigg\{ \int_{{\mathbb{R}}_+} (\text{e}^{-(\alpha x)^{-1} \tau^\alpha |\theta|^\alpha} - 1) x^{\beta-1} \, \text{d} x \bigg\} = {\mathbb{E}} \text{e}^{\text{i} \theta \tau V_{\alpha,\beta}},\quad c \to 0 \end{equation*}

(see (2.8)–(2.10)).

Let $1 = \beta < \alpha$ . For any $\theta \in {\mathbb{R}}$ , we consider ${\mathbb{E}} \text{e}^{\text{i} \theta (\log(1/c) )^{-1/\alpha} c^{-1}\mathcal{Z}_{\alpha,\beta}(c \tau)} = \text{e}^{\sum_{i=1}^3 I_{c,\delta}^{i}}$ , where

\begin{equation*}\sum_{i=1}^3I_{c,\delta}^{i} \coloneqq \bigg\{ \int_0^\delta + \int_{\gfrac{\delta}{c}}^\infty + \int_\delta^{\gfrac{\delta}{c}} \bigg\} ( \text{e}^{-|\theta|^\alpha (\log\pgfrac{1}{c})^{-1} K_c(x) } - 1 ) \, \text{d} x\end{equation*}

with

\begin{equation*} K_c (x) \coloneqq \frac{1}{c^\alpha} \int_{{\mathbb{R}}} |f_{c\tau} (x,s)|^\alpha \, \text{d} s,\end{equation*}

the same as above, given $\delta > 0$ . Then $I_{c,\delta}^1 = {\mathrm{o}} (1)$ , and by (4.3) and (4.29),

\begin{align*}|I^2_{c,\delta}| &\le \frac{C}{c^\alpha \log \pgfrac{1}{c}} \int_{\gfrac{\delta}{c}}^\infty \bigg( \frac{1}{\alpha x} \min \bigg( c \tau, \frac{1}{x} \bigg)^{\alpha} + c \tau \min \bigg( \frac{1}{x}, c\tau \bigg)^\alpha \bigg) \, \text{d} x\\[3pt] &\le \frac{C}{\log \pgfrac{1}{c}} \int_{\delta}^\infty \bigg( \frac{1}{\alpha x} \min \bigg( \tau, \frac{1}{x} \bigg)^{\alpha} + \tau \min \bigg( \frac{1}{x}, \tau \bigg)^\alpha \bigg) \, \text{d} x \\[3pt] & = {\mathrm{o}} (1), \quad c \to 0.\end{align*}

With the notation $\tilde K (cx)\coloneqq x K_c (x)$ , $x \in {\mathbb{R}}_+$ , using (4.3) and $\tilde K (c x) \le C$ , $x>\delta$ , we have

\begin{equation*}I_{c,\delta}^3 \sim -|\theta|^\alpha \int_\delta^{\gfrac{\delta}{c}} \frac{\tilde K(cx)}{x \log \pgfrac{1}{c}} \, \text{d} x, \quad c \to 0,\end{equation*}

for every $\delta >0$ . Moreover,

\begin{equation*}\limsup_{c \to 0} \sup_{\delta <x<\tfrac{\delta}{c}} \bigg|\tilde K(cx) - \frac{\tau^\alpha}{\alpha} \bigg| < \epsilon(\delta)\end{equation*}

with $\lim_{\delta \to 0} \epsilon(\delta) = 0$ , since

\begin{equation*}\tilde K (w) = \frac{1}{\alpha} \bigg( \frac{1-\text{e}^{-w \tau}}{w} \bigg)^\alpha + w \int_0^{\tau} \bigg( \frac{1 - \text{e}^{-w s} }{w} \bigg)^\alpha \, \text{d} s\to \frac{\tau^\alpha}{\alpha}, \quad w \to 0.\end{equation*}

Hence

\begin{equation*}\lim_{\delta \to 0} \limsup_{c \to 0} \bigg|I_{c,\delta}^3 + \frac{\tau^\alpha|\theta|^\alpha}{\alpha}\bigg| = 0,\end{equation*}

finishing the proof of

\begin{equation*}\lim_{c \to 0} {\mathbb{E}} \text{e}^{\text{i} \theta (\log (1/c))^{-1/\alpha} c^{-1} \mathcal{Z}_{\alpha,\beta}(c \tau)} = {\mathbb{E}} \text{e}^{\text{i} \theta \tau V_{\alpha,1}}.\end{equation*}

Finally, consider the large scale limit of $\mathcal{ Z}_{\alpha,\beta}$ as $c \to \infty $ for $0 < \beta < \alpha$ . Then

\begin{equation*} {\mathbb{E}} \text{e}^{\text{i} \theta c^{-1/\alpha} \mathcal{Z}_{\alpha,\beta} (c \tau)} = \exp \bigg\{ \int_{{\mathbb{R}}_+} (\text{e}^{-|\theta|^\alpha K_c (x) } - 1) x^{\beta-1} \, \text{d} x \bigg\}, \end{equation*}

where, using the scaling property,

\begin{align*} K_c (x) &\coloneqq \frac{1}{c} \int_{{\mathbb{R}}} |f_{c\tau} (x,s) |^\alpha \, \text{d} s \\[3pt] & = c^{\alpha} \int_{{\mathbb{R}}} |f_{\tau} (c x,s) |^\alpha \, \text{d} s\\[3pt] &= \frac{(1-\text{e}^{- c x \tau})^\alpha}{c \alpha x^{1+\alpha}} + \frac{1}{x^\alpha} \int_0^{\tau} ( 1 - \text{e}^{-c xs} )^\alpha \, \text{d} s \\[3pt] & \to \frac{\tau}{x^\alpha}, \quad c \to \infty.\end{align*}

It is obvious that, for $x>1$ , $K_c(x)\leq x^{-\alpha}((c\tau)^{-1}+\tau)={\mathrm{O}} (x^{-\alpha})$ . Therefore, by the DCT,

\begin{equation*} {\mathbb{E}} \text{e}^{\text{i} \theta c^{-1/\alpha} \mathcal{Z}(c \tau)} \to \exp \bigg\{ \int_{{\mathbb{R}}_+} ( \text{e}^{- |\theta|^\alpha x^{-\alpha} \tau} - 1 ) x^{\beta-1} \, \text{d} x \bigg\} = {\mathbb{E}} \text{e}^{\text{i} \theta \mathcal{W}_{\alpha,\beta} (\tau)},\quad c \to \infty \end{equation*}

(see (2.8)–(2.10)). This proves part (v) and completes the proof of Proposition 3.1. □