2.1. Non-stationary power-law shot noise processes

As in [Reference Pang and Taqqu35], we consider the shot noise process $S^*=\{S^*(y)\}_{y\in \mathbb R}$ defined by

(1) \begin{equation} S^*(y)\,:\!=\sum_{j\in \mathbb Z_*} g^*(y-S_j)R_j(S_j), \quad y\in \mathbb R,\end{equation}

where $\{S_j\}_{j\in \mathbb Z_*}$ is a sequence of Poisson arrival times with rate 1, satisfying $\cdots < S_{-1} < 0 < S_1 < \cdots$ , and $R_j$ denotes the noise caused by the shot at the time $S_j$ . The non-negative shot shape function $g^*$ accounts for the lingering effect of the noise shocks.

Assume that the random variables $\{R_j\}_{j\in \mathbb Z_*}$ are conditionally independent given $\{S_j\}_{j\in \mathbb Z_*}$ , and that, for each $j \in \mathbb Z_*$ , the distribution of $R_j$ depends on the time $S_j$ . Specifically, for any $u,r \in \mathbb R$ , we have the conditional distribution $\mathbb P(R_j\le r\mid S_j=u)=F_u(r)$ . We write $R_j(S_j)$ to indicate explicitly that $R_j$ depends on $S_j$ .

Condition 1. Given the arrival times $\{S_j\}_{j\in \mathbb Z_*}$ , we assume that the conditional mean of $R_j$ is zero, and the conditional variance of $R_j$ is finite. Specifically, for each $u \in \mathbb R\setminus\{0\}$ ,

(2) \begin{equation} K_1(u)\,:\!=\int_{\mathbb R}r\,{\mathrm{d}} F_u(r)=0, \qquad K_2(u)\,:\!=\int_{\mathbb R}r^2\,{\mathrm{d}} F_u(r)\in(0,\infty). \end{equation}

Furthermore, we assume that $K_2(u)=|u|^{-\gamma}\tilde L(u)\mathbf{1}_{\{u\neq0\}}$ , where $\gamma\in(0,1)$ is a constant, $\tilde L(u)$ is a positive slowly varying function at $\pm\infty$ , and $\tilde L(u)$ is bounded away from 0 and $\infty$ on every compact subset of $\mathbb R$ .

Consider the integrated shot noise process $S\,:\!=\{S(t)\}_{t\in \mathbb R_+}$ defined by

(3) \begin{equation} S(t)\,:\!=\sum_{j\in \mathbb Z_*}\big(g(t-S_j)-g(\!-S_j)\big) R_j(S_j), \quad t\in \mathbb R_+,\end{equation}

where $S(0)=0$ , and the shot shape function g(t) is given by

(4) \begin{equation} g(t)\,:\!=\int_0^t g^*(y)\,{\mathrm{d}} y, \quad t\in \mathbb R_+.\end{equation}

We assume that the function $g^*$ satisfies the following condition.

Condition 2. There exists a constant $\beta\in\big(\frac12-{\gamma}/{2}, 1\big)$ such that

(5) \begin{equation} g^*(y)\,:\!=y^{-\beta}L^*(y) \mathbf{1}_{\{y>0\}}, \end{equation}

where $L^*$ is a positive slowly varying function at $+\infty$ that is bounded away from 0 and $\infty$ on every bounded subset of $[0,\infty)$ .

Under Condition 2, by using [Reference Bingham, Goldie and Teugels5, Proposition 1.5.8] we know that the function g in (4) satisfies $g(t)=t^{1-\beta}L(t)\mathbf{1}_{\{t\ge0\}}$ , where L is a positive slowly varying function bounded away from 0 and $\infty$ on every bounded subset of $[0,\infty)$ , and $L(t) \sim {L^*(t)}/({1-\beta})$ as $t\rightarrow\infty$ . Let N be a Poisson random measure on $(\mathbb R^2, \mathcal B(\mathbb R^2))$ with intensity ${\mathrm{d}} F_u(r)\,{\mathrm{d}} u$ , that is,

1. (i) N is independently scattered on $(\mathbb R^2, \mathcal B(\mathbb R^2))$ , i.e. for any disjoint sets $G_1,\ldots,G_n$ , $N(G_1),\ldots,N(G_n)$ are independent;

2. (ii) for any bounded Borel set $G \in \mathcal B(\mathbb R^2)$ , N(G) is a Poisson random variable with mean $\int_{G}{\mathrm{d}} F_u(r)\,{\mathrm{d}} u$ .

Let $\widetilde N$ be the compensated Poisson measure defined by $\widetilde N(G)=N(G)-\int_G {\mathrm{d}} F_u(r)\,{\mathrm{d}} u$ for any bounded Borel set $G \in \mathcal B(\mathbb R^2)$ . Since the conditional mean of $R_j$ in (2) is zero, the integrated shot noise process $\{S(t)\}_{t\in \mathbb R_+}$ can be written as

\begin{equation*} S(t) = \int_{-\infty}^t\int_{-\infty}^{\infty}\big(g(t-u)-g(\!-u)\big)r\widetilde N({\mathrm{d}} u,{\mathrm{d}} r), \quad t\ge 0.\end{equation*}

The integral on the right-hand-side is well defined in the sense of [Reference Rajput and Rosinski42], since

(6) \begin{align} \mathbb E[S(t)^2] & = \int_{-\infty}^t\int_{-\infty}^{\infty}\big(g(t-u)-g(\!-u)\big)^2 r^2\,{\mathrm{d}} F_u(r)\,{\mathrm{d}} u \nonumber \\ & = \int_{-\infty}^t\big(g(t-u)-g(\!-u)\big)^2 K_2(u)\,{\mathrm{d}} u, \end{align}

which is finite under Conditions 1 and 2. We also refer to [Reference Pipiras and Taqqu39, Appendix B.1.4] for more information on integration with respect to Poisson measures.

2.2. Weak convergence of the scaled integrated shot noise process

Condition 3. Let $b\colon\mathbb N_*\rightarrow\mathbb R_+$ be a function that satisfies

\begin{equation*} \lim_{n\rightarrow\infty}\frac{b(n)}{n^{{3}/{2}-\beta-{\gamma}/{2}}L(n)\tilde L(n)^{1/2}} = 1. \end{equation*}

When $\beta\in\big(\frac12,1\big)$ and $\gamma\in(0,1)$ , under Conditions 1–3 and the fourth moment condition of the distribution $F_u$ (see (9)), [Reference Pang and Taqqu35, Theorem 3.1 and Remark 3.3] proved that, as $ n\rightarrow\infty$ , $\{{S(nt)}/{b(n)}\}_{t\in [0,T]}$ converges weakly to a generalized fractional Brownian motion (GFBM) X in $(\mathcal D[0,T], J_1)$ , where $X\,:\!=\{X(t)\}_{t\in [0,T]}$ is defined by the integral representation

\begin{equation*} X(t) =\int_{\mathbb R}\big((t-u)_+^{1-\beta}-(\!-u)_+^{1-\beta} \big)|u|^{-{\gamma}/{2}}\,B({\mathrm{d}} u),\end{equation*}

with $u_+ = \max\{0, u\}$ for all $u \in \mathbb R$ , and $B({\mathrm{d}} u)$ a Gaussian random measure on $\mathbb R$ with the Lebesgue control measure ${\mathrm{d}} u$ . The process X is a continuous mean-zero Gaussian process with $X(0)=0$ , and it has the self-similarity property with Hurst parameter $H=\frac32-\beta-{\gamma}/{2}$ . When $\gamma=0$ , X becomes a standard FBM with Hurst parameter H; when $\gamma\in (0,1)$ , X is a GFBM as defined in [Reference Pang and Taqqu35]. A GFBM preserves the self-similarity property, while the factor $|u|^{-{\gamma}/{2}}$ introduces the non-stationarity of increments, making it suitable for modeling non-stationary physical systems. For example, [Reference Ichiba, Pang and Taqqu21] studied the semimartingale properties of GFBMs, and applied them to model volatility processes in finance.

It follows from [Reference Ichiba, Pang and Taqqu20, Theorem 3.1] that for any $\gamma\in(0,1)$ , $\beta\in\big(\frac12-{\gamma}/{2},1\big)$ , there exists a constant $c_{2,1}(\gamma, \beta)\in (0,\infty)$ such that, for any $t, s\ge0$ ,

(7) \begin{equation} \mathbb E[|X(t)-X(s)|^2] = \int_{\mathbb R}\big((t-u)_+^{1-\beta}-(s-u)_+^{1-\beta}\big)^2|u|^{-\gamma}{\mathrm{d}} u \le c_{2,1}(\gamma,\beta)|t-s|^{3-2\beta-\gamma}.\end{equation}

The following theorem proves the convergence of $\{S(nt)/b(n)\colon t\in[0,T]\}_{n\in\mathbb N_*}$ in the space $\big(\mathcal C[0,T], \|\cdot\|_T\big)$ .

2.3. Sample path MDP

To establish the sample path MDP for the integrated shot noise processes with Poisson arrivals and non-stationary noise, we impose the following conditions.

Condition 4. There are constants $u_0>0$ and $\theta_0>0$ such that $K_2(u)=|u|^{-\gamma}\tilde L(u)\mathbf{1}_{\{|u|\ge u_0\}}$ , where $\tilde L(u)$ is a positive slowly varying function at $\pm\infty$ that it is bounded away from 0 and $\infty$ on every compact subset of $\mathbb R$ , and

(8) \begin{equation} M_{\theta_0} \,:\!= {\sup}_{u\in\mathbb R}\int_{\mathbb R}\exp\!\big(\theta_0|r|\big)\,{\mathrm{d}} F_u(r) < +\infty. \end{equation}

Condition 5. There exists a positive constant $\kappa < \infty$ such that, for all $|u|\ge u_0$ ,

(9) \begin{equation} \frac{K_4(u)}{K_2(u)^2}\le \kappa, \end{equation}

where $K_4(u) = \int_{\mathbb R}r^4\,{\mathrm{d}} F_u(r)$ .

Condition 6. Assume that the function $\pi\colon\mathbb R_+\rightarrow\mathbb R_+$ satisfies the following conditions:

\begin{equation*} \lim_{n\rightarrow\infty}\pi(n) = \infty, \qquad \lim_{n\rightarrow\infty}\frac{\pi(n)}{n^{1/2-{\gamma}/{2}}\tilde L(n)^{1/2}} = 0. \end{equation*}

For any $n>0$ and $t\ge0$ , we define

(10) \begin{equation} S_n(t)\,:\!= \frac{S(nt)}{b(n)\pi(n)}.\end{equation}

Theorem 2. Assume Conditions 1–6 hold with $\gamma\in(0,1)$ and $\beta\in\big(\frac12-{\gamma}/{2},1\big)$ . Then, as $n\rightarrow\infty$ , $\{S_n(t)\colon t\in[0,T]\}_{n\in \mathbb N_*}$ satisfies the large deviation principle on $\big(\mathcal C[0,T],\|\cdot\|_{T}\big)$ with the speed function $\pi(n)^2$ and the good rate function $I_{\beta,\gamma}$ given by

(11) \begin{equation} I_{\beta,\gamma}(\xi) \,:\!= \inf_{\mathcal S_{\xi}}\frac{1}{2} \int_{-\infty}^T\psi(u)^2\,{\mathrm{d}} u, \end{equation}

where, for $\xi\in\mathcal C[0,T]$ , $\mathcal S_{\xi}$ is the collection of all $\psi\in L^2((\!-\infty,T],\mathbb R)$ such that

$$ \xi(t) = \int_{-\infty}^t\big((t-u)_+^{1-\beta}-(\!-u)_+^{1-\beta} \big)|u|^{-{\gamma}/{2}}\psi(u)\,{\mathrm{d}} u \quad \text{for } t\in [0,T]. $$

To explain why the distribution $F_u$ is assumed to be degenerate when u is in a neighborhood of the origin, let us recall that, when applying the Gärtner–Ellis theorem [Reference Dembo and Zeitouni12, Section 2.3], it is usually required that, for every $\theta$ in a neighborhood of the origin,

(12) \begin{equation} \Lambda(\theta) \,:\!= \lim_{n\rightarrow\infty}\frac{1}{\pi(n)^2}\log\mathbb E[\exp\!(\theta\pi(n)^2S_n(t))] \ \text{exists as a real number.}\end{equation}

For the MDP of the empirical mean of independent and identically distributed random variables, Condition (12) reduces to the well-known Cramér condition; see, e.g., [Reference Dembo and Zeitouni12, Theorem 3.7.1] or [Reference Gao17, Theorem 1].

Let us continue with Example 1. If $F_u$ is Gaussian with mean zero and variance $K_2(u) = |u|^{-\gamma}\mathbf{1}_{\{|u|\neq 0\}}$ for $\gamma \in (0,1)$ then, for every $\theta>0$ , $\mathbb E[\!\exp(\theta\pi(n)^2S_n(t))] = \infty$ , which contradicts (12). Thus, to study the MDP, it is necessary to assume that $F_u$ is degenerate when $|u|< u_0$ for some $u_0>0$ , which is equivalent to saying that there is no noise when the shot time $|S_j|<u_0$ for every $j\in \mathbb Z_*$ . Under this condition, the integrated shot noise process $\{S(t)\}_{t\in[0,T]}$ defined by (3) becomes the following truncated model:

(13) \begin{equation} \tilde S(t) \,:\!= \sum_{j\in\mathbb Z_*}\big(g(t-S_j)-g(\!-S_j)\big)R_j(S_j)\mathbf{1}_{\{|S_{j}|\ge u_0\}}, \quad t\in\mathbb R_+.\end{equation}

Since $S_j\rightarrow+\infty$ as $|j|\rightarrow\infty$ , the truncation can only affect a finite number of terms in (13). Therefore, $\{\tilde S(nt)/b(n)\}_{t\in [0,T]}$ shares many properties of $\{S(nt)/b(n)\}_{t\in [0,T]}$ . In particular, Theorem 1 also holds for $\{\tilde S(nt)/b(n)\colon t\in[0,T]\}_{n\in \mathbb N_*}$ under Conditions 1–3.

2.4. Applications

Shot noise processes and integrated shot noise processes have been applied to model the claim amount processes in insurance risk theory or input processes in queueing theory [Reference Brémaud8, Reference Ganesh and Torrisi16, Reference Klüppelberg and Mikosch27, Reference Klüppelberg, Mikosch and Schärf28, Reference Pang and Taqqu35, Reference Whitt50]. Those models are also applied to study the dynamics in storage systems or dams [Reference Prabhu40].

The process $\{S(t)\}_{t\ge0}$ in the context of insurance can be interpreted as follows. Consider an insurance company with initial capital $M_{0}>0$ , premium rate $\nu_{1}>0$ , and insurance rate $\nu_2\in(0,\nu_1)$ . Suppose that claims occur according to a renewal process $\{S_j\}_{j\in \mathbb Z_*}$ , and that the insurance company pays claims occurring at time $\{S_j\}_{j\in \mathbb Z_*}$ , at the rate $g^*(\!\cdot-S_j) R_j(S_j)+\nu_{2} $ , with $g^*(\!\cdot\!)$ given in (1). Then, the total claim paid in the time interval (0,t] is the integrated shot noise process $\{S(t)+\nu_2 t\}_{t\ge0}$ , with S(t) defined in (3). See [Reference Ganesh and Torrisi16] for further details.

Write $\nu=\nu_{1}-\nu_{2}$ . For any $t\ge0$ , let

(14) \begin{equation} M(t) \,:\!= {\sup}_{0\le s\le t}(S(s)-\nu s); \qquad V(t) \,:\!= {\sup}_{0\le s\le t}(S(t)-S(s)-\nu(t-s)). \end{equation}

Then, $M(t)>0$ represents the event of the insurance company’s ruin in the time interval [0,t], and V(t) is the total amount of residual claims, which are paid from the claims occurring up to time t. In the language of queueing models, $\{S(t)+\nu_2t\}_{t\ge0}$ is called an input process, $\{M(t)\}_{t\ge0}$ is called a running maximum process, and $\{V(t)\}_{t\ge0}$ is called a workload process [Reference Prabhu40, Reference Whitt50].

A systematic method for generating an input process that exhibits long-range dependence was developed in [Reference Chang, Yao and Zajic10, Reference Chang, Yao and Zajic11],] which showed that $V^{n}=\{V(nt)/t\colon t\ge0\}_{n\in\mathbb N_*}$ and $M^{n}=\{M(nt)/t\colon t\ge0\}_{n\in\mathbb N_*}$ satisfy the MDP [Reference Chang, Yao and Zajic10, Theorem 4.1, Lemma 5.2] and that the limit $M(\infty)$ has a sub-exponential tail distribution [Reference Chang, Yao and Zajic10, Theorem 5.1]. On the other hand, [Reference Anugu and Pang1] studied the LDP for the stochastic fluid queue with the generalized Riemann–Liouville FBM as the input process, examining the long-time behavior of the processes $\{M(t)\}_{t\ge0}$ and $\{V(t)\}_{t\ge0}$ , as defined by (14), with S replaced by the generalized Riemann–Liouville FBM. The methods there rely on large deviation theory and some special properties of the generalized Riemann-Liouville FBM.

We are interested in the asymptotic probabilities of the following events associated with the integrated shot noise process, as $t\rightarrow\infty$ :

1. (i) the ruin probability of the insurance company in the time interval [0, t], i.e.

   $$ \mathbb P\big({\sup}_{0\le s\le t}(S(s) - \nu s) > M_{0}\big); $$

2. (ii) the probability that the total amount of residual claims paid from the time of the claim to time t exceeds a given value $M_{1}$ , i.e.

   $$ \mathbb P\big({\sup}_{0\le s\le t}(S(t)-S(s)-\nu(t-s)) > M_{1}\big). $$

Next, we give estimates for these probabilities using the MDP in Theorem 2. Let $c_{\beta,\gamma} = \int_{\mathbb R}\big((1-u)_+^{1-\beta}-(-u)_+^{1-\beta}\big)^2|u|^{-\gamma}\,{\mathrm{d}} u$ .

Proposition 2. Under the conditions of Proposition 1, M(t) converges almost surely to a finite random variable $M(\infty)$ , which has a sub-exponential tail distribution. Specifically,

$$ \lim_{n\rightarrow\infty}\frac{b(n)^2}{n^2}\log\mathbb P(M(\infty)>n) = -\frac{\nu^{2H}}{2c_{\beta,\gamma}H^{2H}(1-H)^{2(1-H)}}. $$

2.5. The shot noise process with positive Poisson arrival times of shots

Consider the integrated shot noise process $\widetilde S\,:\!=\{\widetilde S(t)\}_{t\in \mathbb R_+}$ defined by

\begin{equation*} \widetilde S(t) \,:\!= \sum_{j\in\mathbb N_*}\big(g(t-S_j)-g(\!-S_j)\big)R_j(S_j), \quad t\in \mathbb R_+.\end{equation*}

Here, the processes g, $\{S_j\}_{j\in \mathbb N_*}$ , and $\{R_j\}_{j\in \mathbb N_*}$ satisfy Conditions 1 and 2 with $\gamma\in(0,1)$ and $\beta\in\big(\frac12-{\gamma}/{2},1\big)$ .

When b(n) satisfies Condition 3, by following similar steps to the proof of Theorem 1, we can show that $\{{\widetilde S(nt)}/{b(n)}\}_{t\in [0,T]}$ converges weakly to $\{Z(t)\}_{t\in[0,T]}$ in $\big(\mathcal C[0,T],\|\cdot\|_T\big)$ as $n\rightarrow \infty$ . Here, $\{Z(t)\}_{t \ge 0}$ is the generalized Riemann–Liouville FBM defined by

(17) \begin{equation} Z(t)\,:\!=\int_0^t(t-u)^{1-\beta}u^{-{\gamma}/{2}}B({\mathrm{d}} u), \quad t\ge0,\end{equation}

where $B({\mathrm{d}} u)$ is a Gaussian measure. This process was introduced in [Reference Pang and Taqqu35, Remark 5.1] and further studied in [Reference Ichiba, Pang and Taqqu20, Section 2.2] and [Reference Wang and Xiao49, Section 3].

For every $n\in\mathbb N_*$ and $t\in[0,T]$ , let $\widetilde S_n(t)$ be defined as in (10) with S(nt) replaced by $\widetilde S(nt)$ . By following the same reasoning as in the proof of Theorem 2 we obtain that, under the same conditions as Theorem 2, $\{\widetilde S_n(t)\colon t\in[0,T]\}_{n\in\mathbb N_*}$ satisfies the LDP on $\big(\mathcal C[0,T], \|\cdot\|_{T}\big)$ as $n\rightarrow\infty$ , with speed function $\pi(n)^2$ and good rate function $\widetilde I_{\beta,\gamma}$ given by

(18) \begin{equation} \widetilde I_{\beta,\gamma}(\xi) \,:\!= \inf_{\psi\in\widetilde{\mathcal S}_{\xi}}\frac{1}{2}\int^T_{0}\psi(u)^2\,{\mathrm{d}} u,\end{equation}

where, for $\xi\in\mathcal C[0,T]$ , $\widetilde{\mathcal S}_{\xi}$ is the collection of all $\psi\in L^2[0,T]$ such that

$$\xi(t) = \int_{0}^t\big((t-u)_+^{1-\beta}-(\!-u)_+^{1-\beta}\big)|u|^{-{\gamma}/{2}}\psi(u)\,{\mathrm{d}} u \quad\text{for } t\in[0,T].$$

Notice that the rate function $\widetilde I_{\beta,\gamma}(\xi)$ in (18) is exactly the rate function for LDP of the generalized Riemann–Liouville FBM; see [Reference Anugu and Pang1, Theorem 3.1].

As an application of the LDP for $\{\widetilde S_n\}_{n\ge1}$ , following the steps in Section 2.4, we can also obtain similar results for $\widetilde M$ and $\widetilde V$ , where $\widetilde M(t) \,:\!= {\sup}_{0\le s\le t}(\widetilde S(s)-\nu s)$ and $\widetilde V(t) \,:\!= {\sup}_{0\le s\le t}(\widetilde S(t)-\widetilde S(s)-\nu(t-s))$ .

2.6. Results without Condition 5

When Condition 5 is absent, by using a strategy similar to the one in the previous subsections we can prove the same conclusion as Theorem 2, provided that the function $\pi(n)$ has a slower rate as n tends to infinity. More precisely, we assume $\pi(n)$ satisfies the following condition.

Condition 7. Assume that there exists some constant $\eta>0$ satisfying $\lim_{n\rightarrow\infty} \pi(n)= \infty$ and

\begin{equation*} \lim_{n\rightarrow\infty} \frac{\pi(n)}{n^{1/2-{\gamma}/{2}-\eta}}=0. \end{equation*}

3.1. Two lemmas

First, we provide a uniform moment estimate for the increments of $\hat S_n$ .

Lemma 1. Assume that Conditions 1–3 hold with $\gamma\in(0,1)$ and $\beta\in\big(\frac12-{\gamma}/{2}, 1-{\gamma}/{2}\big)$ . Then, for any $0 < \eta < \min\{2\beta+\gamma-1,2-2\beta-\gamma\}$ there exist positive constants $c_{3,1}$ and $c_{3,2}$ such that, for any $n>c_{3,2}$ ,

(19) \begin{equation} \mathbb E\big[|\hat S_n(t)-\hat S_n(s)|^2\big] \le c_{3,1}|t-s|^{3-2\beta-\gamma-\eta} \quad \text{for all } s,t\in [0,T] . \end{equation}

Consequently, by Kolmogorov’s continuity theorem (see, e.g., [Reference Khoshnevisan24, Theorem C.6]), $\hat S_n$ has a modification, still denoted by $\hat S_n$ , that is almost surely continuous on [0, T].

Proof. For any $n>0$ , $0\le s < t\le T$ , we write

\begin{align*} \hat S_n(t)-\hat S_n(s) & = \frac{1}{b(n)}\int_{-\infty}^{ns}\int_{-\infty}^{\infty}\big(g(nt-u)-g(ns-u)\big)r\,N({\mathrm{d}} u,{\mathrm{d}} r) \\ & \quad + \frac{1}{b(n)}\int_{ns}^{nt}\int_{-\infty}^{\infty}g(nt-u)r\,N({\mathrm{d}} u,{\mathrm{d}} r). \end{align*}

Since the two terms on the right-hand side are independent and have expectation zero, we have

(20) \begin{align} \mathbb E\big[(\hat S_n(t)-\hat S_n(s))^2\big] & = \frac{1}{b(n)^2}\int_{-\infty}^{ns}\big(g(nt-u)-g(ns-u)\big)^2K_2(u)\,{\mathrm{d}} u \nonumber\\ & \quad + \frac{1}{b(n)^2}\int_{ns}^{nt}g(nt-u)^2K_2(u)\,{\mathrm{d}} u \nonumber \\ & = \frac{n}{b(n)^2}\int_{-\infty}^{s}\big(g(nt-nu)-g(ns-nu)\big)^2K_2(nu)\,{\mathrm{d}} u \nonumber \\ & \quad + \frac{n}{b(n)^2}\int_{s}^{t}g(nt-nu)^2K_2(nu)\,{\mathrm{d}} u, \end{align}

where a change of variables is used in the last step.

Since the function $L^*$ is slowly varying at $\infty$ , it follows from Potter’s theorem [Reference Bingham, Goldie and Teugels5, Theorem 1.5.6] that, for any $\delta_1>0$ , there exists a constant $c_{3,3}(\delta_1)>1$ such that the function $L^*$ in (5) satisfies

(21) \begin{equation} \bigg|\frac{L^*(u)}{L^*(v)}\bigg| \le c_{3,3}(\delta_1)\max\bigg\{\bigg(\frac{u}{v}\bigg)^{\delta_1}, \bigg(\frac{u}{v}\bigg)^{-\delta_1}\bigg\} \quad \text{for all } u,v>0. \end{equation}

By (4), (5), and (21), for any $\delta_1\in (0,1-\beta)$ ,

(22) \begin{align} & \frac{|g(nt-nu)-g(ns-nu)|}{n^{1-\beta}L^*(n)} \nonumber \\ & \quad = \int_{(s-u)_+}^{(t-u)_+}y^{-\beta}\frac{L^*(ny)}{L^*(n)}\,{\mathrm{d}} y \nonumber \\ & \quad \le c_{3,3}(\delta_1)\max\bigg\{\int_{(s-u)_+}^{(t-u)_+}y^{-\beta+\delta_1}\,{\mathrm{d}} y, \int_{(s-u)_+}^{(t-u)_+}y^{-\beta-\delta_1}\,{\mathrm{d}} y\bigg\} \nonumber \\ & \quad = c_{3,3}(\delta_1)\max\bigg\{\frac{(t-u)_+^{1-\beta+\delta_1}-(s-u)_+^{1-\beta+\delta_1}} {1-\beta+\delta_1}, \frac{(t-u)_+^{1-\beta-\delta_1}-(s-u)_+^{1-\beta-\delta_1}}{1-\beta-\delta_1}\bigg\}. \end{align}

Applying Potter’s theorem to the function $\tilde L$ given in Condition 1, we obtain that for any $\delta_2>0$ there exists a constant $c_{3,4}(\delta_2)>0$ such that ${\tilde L(nu)}/{\tilde L(n)} \le c_{3,4}(\delta_2)\max\{|u|^{\delta_2},|u|^{-\delta_2}\}$ for any $u\neq 0$ . Hence, from this, Condition 3, and (22), we derive that for any $\delta_1\in (0,1-\beta)$ and $\delta_2>0$ there exist constants $c_{3,5}>0$ and $c_{3,6}=c_{3,6}(\delta_1,\delta_2,\beta)>0$ such that, for any $n\ge c_{3,5}$ ,

(23) \begin{align} & \frac{n}{b(n)^2}\big(g(nt-nu)-g(ns-nu) \big)^2K_2(nu) \nonumber \\ & \quad = \frac{n^{3-2\beta}L^*(n)^2\tilde L(n)}{b(n)^2}\bigg(\frac{g(nt-nu)-g(ns-nu)}{n^{(1-\beta)}L^*(n)}\bigg)^2 \bigg(|nu|^{-\gamma}\frac{\tilde L(nu)}{\tilde L(n)}\bigg) \nonumber \\ & \quad \le c_{3,6}\max\big\{\big((t-u)_+^{1-\beta+\delta_1}-(s-u)_+^{1-\beta+\delta_1}\big)^2, \big((t-u)_+^{1-\beta-\delta_1}-(s-u)_+^{1-\beta-\delta_1}\big)^2\big\} \nonumber \\ & \qquad \times \max\{|u|^{\delta_2-\gamma},|u|^{-\delta_2-\gamma}\}. \end{align}

By (7) and (23), we obtain that for any $s,t\in[0,T]$ , $n\ge c_{3,5}$ , and positive constants $\delta_1<1-\beta$ , $\delta_2<\min\{\gamma,1-\gamma\}$ with $\delta_1+{\delta_2}/{2} \le \min\big\{\beta+{\gamma}/{2}-\frac12,1-\beta-{\gamma}/{2}\big\}$ ,

(24) \begin{align} & \frac{n}{b(n)^2}\int_{-\infty}^{s}\big(g(nt-nu)-g(ns-nu)\big)^2K_2(nu)\,{\mathrm{d}} u \nonumber \\ & \quad \le c_{3,6}(\delta_1,\delta_2,\beta)\bigg[ \int_{-\infty}^s\big[(t-u)_+^{1-\beta+\delta_1}-(s-u)_+^{1-\beta+\delta_1}\big]^2 (|u|^{\delta_2-\gamma}+|u|^{-\delta_2-\gamma})\,{\mathrm{d}} u \nonumber \\ & \qquad\qquad\qquad\qquad + \int_{-\infty}^s\big[(t-u)_+^{1-\beta-\delta_1}-(s-u)_+^{1-\beta-\delta_1}\big]^2 (|u|^{\delta_2-\gamma} + |u|^{-\delta_2-\gamma})\,{\mathrm{d}} u\bigg] \nonumber \\ & \quad \le c_{3,7}(\delta_1,\delta_2,\beta,\gamma)\bigg[ |t-s|^{3-2\beta-\gamma+2\delta_1+\delta_2 }+|t-s|^{3-2\beta-\gamma+2\delta_1-\delta_2 } \nonumber \\ & \quad\qquad\qquad\qquad\qquad + |t-s|^{3-2\beta-\gamma-2\delta_1+\delta_2 } + |t-s|^{3-2\beta-\gamma-2\delta_1-\delta_2 }\bigg] \nonumber \\ & \quad\le c_{3,7}(\delta_1,\delta_2,\beta,\gamma)T^{4\delta_1+2\delta_2}|t-s|^{3-2\beta-\gamma-2\delta_1-\delta_2}, \end{align}

where $c_{3,7}(\delta_1,\delta_2,\beta,\gamma)$ is a finite constant.

Similarly to the proof of (24), there exists a constant $c_{3,8}>0$ such that, for all $n\ge c_{3,8}$ , $s,t\in[0,T]$ , and positive constants $\delta_1<1-\beta$ , $\delta_2<\min\{\gamma,1-\gamma\}$ with $\delta_1 + {\delta_2}/{2} \le \min\big\{\beta+{\gamma}/{2}-\frac12,1-\beta-{\gamma}/{2}\}$ , we have

(25) \begin{equation} \frac{1}{b(n)^2}\int_{s}^{t}g(nt-nu)^2K_2(nu)\,{\mathrm{d}} u \leq c_{3,9}(\delta_1,\delta_2,\beta,\gamma)T^{4\delta_1+2\delta_2}|t-s|^{3-2\beta-\gamma-2\delta_1-\delta_2}. \end{equation}

Here, $c_{3,9}(\delta_1,\delta_2,\beta,\gamma)$ is a finite constant.

Putting (20), (24), and (25) together, we obtain (19) with $\eta=2\delta_1+\delta_2$ . The proof is complete.

3.2. Proof of Theorem 1

In order to verify the (exponential) tightness in the proofs of Theorems 1 and 2, we will make use of the following Hölder norm. For any $\tau\in (0,1)$ ,

$$\|\xi\|_{T,\tau} \,:\!= {\sup}\bigg\{\frac{|\xi(s)-\xi(t)|}{|s-t|^{\tau}}\colon s,t\in[0,T],s\neq t\bigg\}.$$

Let $\mathcal C^{\tau}[0,T]$ be the space of all $\tau$ -Hölder continuous functions, equipped with the $\|\cdot\|_{T,\tau}$ norm.

Proof of Theorem 1. The convergence of finite-dimensional distributions was proved in [Reference Pang and Taqqu35, Theorem 3.1]. Specifically, for any integer $k \ge 1$ and $t_i\ge 0$ , $i=1,\ldots,k$ , $(\hat S_n(t_1),\ldots,\hat S_n(t_k))$ converges weakly to $(X(t_1),\ldots,X(t_k))$ in $\mathbb R^k$ as $n\rightarrow\infty$ . To prove Theorem 1, it remains to establish the tightness of $\{\hat S_n(t)\colon t\in[0,T]\}_{n\in\mathbb N_*}$ in the space $\big(\mathcal C[0,T],\|\cdot\|_{T}\big)$ .

Using Lemma 1 and Kolmogorov’s continuity theorem (see, e.g., [Reference Khoshnevisan24, Theorem C.6]), there exist constants $\tau>0$ , $c_{3,10}>0$ , and $c_{3,11}>0$ such that, for all $n\ge c_{3,10}$ , $\mathbb E\big(\|\hat S_n\|_{T,\tau}^2\big) \le c_{3,11}$ . It follows from Chebyshev’s inequality that, for any $M>0$ , $\mathbb P\big(\|\hat S_n\|_{T,\tau} \ge M\big) \le {c_{3,11}}/{M^2}$ , which implies the tightness of $\{\hat S_n\}_{n\in\mathbb N_*}$ in $\big(\mathcal C[0,T],\|\cdot\|_{T}\big)$ , since the Hölder space $\mathcal C^{\tau}[0,T]$ is compactly embedded in $\mathcal C[0,T]$ . The proof is complete.

4.1. MDP of the finite-dimensional distributions

We first give the following formula for the exponential moment of S(t).

Lemma 2. Under the conditions of Theorem 2, for any $T>0$ there exists a constant $\theta_1>0$ such that, for any $|\theta|\le\theta_1$ and $t\in [0,T]$ ,

(26) \begin{equation} \mathbb E\big[{\mathrm{e}}^{\theta S(t)}\big] = \exp\!\bigg(\int_{-\infty}^{t}\int_{-\infty}^{\infty}\bigg[ \exp\!\big(\theta\big(g(t-u)-g(\!-u)\big)r\big) - 1\bigg]\,{\mathrm{d}} F_u(r)\,{\mathrm{d}} u\bigg) < \infty. \end{equation}

Proof. Let $G(t,u, r) \,:\!= \big(g(t-u)-g(\!-u)\big)r$ . For any $M>0$ , let

$$G_{(M)}(t,u,r) \,:\!= G(t,u,r)\cdot\mathbf{1}_{\{u\in[-M,t],|r|\le M\}},$$

and let $S_{(M)}(t) \,:\!= \int_{-\infty}^t\int_{-\infty}^{\infty}G_{(M)}(t,u,r)\,N({\mathrm{d}} u,{\mathrm{d}} r)$ , $t\ge 0$ . By Condition 1 and (6), $S_{(M)}(t)\rightarrow S(t)$ in $L^2(\Omega,\mathcal F,\mathbb P)$ and $G_{(M)}(t,\cdot,\cdot)\in L^1(\mathbb R^2,{\mathrm{d}} F_u(r)\,{\mathrm{d}} u)$ . Consequently, by Campbell’s theorem [Reference Kingman25, p. 28],

(27) \begin{equation} \log\mathbb E\big[\exp\!\big(\theta S_{(M)}(t)\big)\big] = \int_{-\infty}^t\int_{-\infty}^{\infty}\big[\exp\!(\theta G_{(M)}(t,u,r))-1\big]\,{\mathrm{d}} F_u(r)\,{\mathrm{d}} u \end{equation}

for any $\theta\in\mathbb R$ for which the integral on the right-hand side of (27) converges.

We claim that there exists a constant $\theta_1>0$ such that, for any $|\theta|\le \theta_1$ ,

(28) \begin{equation} \int_{-\infty}^t\int_{-\infty}^{\infty}[\!\exp\!(\theta G(t,u,r)) - 1]\,{\mathrm{d}} F_u(r)\,{\mathrm{d}} u < \infty. \end{equation}

We postpone the proof of (28). Let us first prove (26).

Notice that $G_{(M)}(t,u,r)\rightarrow G(t,u,r)$ and $|G_{(M)}(t,u,r)|\le|G(t,u,r)|$ for any $t\ge0$ and $(u,r)\in\mathbb R^2$ . Thus, by (27), (28), and the dominated convergence theorem, we obtain (26).

Now we prove (28). For any $t\in [0,T]$ and $u\in\mathbb R$ , by (2),

(29) \begin{multline} \int_{-\infty}^{\infty}\big[\exp\!\big(\theta\big(g(t-u)-g(\!-u)\big)r\big) - 1\big]\,{\mathrm{d}} F_u(r) \\ = \int_{-\infty}^{\infty}\big[\exp\!\big(\theta\big(g(t-u)-g(\!-u)\big)r\big) - 1 - \theta \big(g(t-u)-g(\!-u)\big)r\big]\,{\mathrm{d}} F_u(r). \end{multline}

By using the elementary inequality

(30) \begin{equation} \big|{\mathrm{e}}^x-1-x\big| \le x^2{\mathrm{e}}^{|x|} \quad \text{for all } x \in \mathbb R, \end{equation}

and by using the Cauchy–Schwarz inequality, we obtain

(31) \begin{align} & \int_{-\infty}^{\infty}\big[\exp\!\big(\theta\big(g(t-u)-g(\!-u)\big)r\big) - 1 - \theta\big(g(t-u)-g(\!-u)\big)r\big]\,{\mathrm{d}} F_{u}(r) \nonumber \\ & \quad \le \theta^2\big(g(t-u)-g(\!-u)\big)^2\int_{-\infty}^{\infty}|r|^2 \exp\!\big(\theta\big|\big(g(t-u)-g(\!-u)\big)r\big|\big)\,{\mathrm{d}} F_{u}(r) \nonumber \\ & \quad \le \theta^2\big(g(t-u)-g(\!-u)\big)^2\bigg(\int_{-\infty}^{\infty}|r|^4\, {\mathrm{d}} F_{u}(r)\bigg)^{{1}/{2}} \nonumber \\ & \qquad \times \bigg(\int_{-\infty}^{\infty}\exp\!\big(2\theta\big|\big(g(t-u)-g(\!-u)\big)r\big|\big)\, {\mathrm{d}} F_{u}(r)\bigg)^{{1}/{2}}. \end{align}

By (4), we know that ${\sup}_{t\in[0,T],u\in\mathbb R}|g(t-u)-g(\!-u)| \le {\sup}_{t\in[0,T]}g(t) =: c_{g,T} < \infty$ . By using Condition 4, for any $\theta\le\theta_0/(2c_{g,T})$ and $u\in\mathbb R$ ,

(32) \begin{equation} \int_{-\infty}^{\infty}\exp\!\big(2\theta|\big(g(t-u)-g(\!-u)\big)r|\big)\,{\mathrm{d}} F_{u}(r) \le \int_{-\infty}^{\infty}\exp\!(\theta_0|r|)\,{\mathrm{d}} F_{u}(r) \le M_{\theta_0}. \end{equation}

By (9), we obtain

(33) \begin{equation} \int_{-\infty}^t\big(g(t-u)-g(\!-u)\big)^2\bigg(\int_{-\infty}^{\infty}|r|^4\,{\mathrm{d}} F_{u}(r)\bigg)^{{1}/{2}}\,{\mathrm{d}} u \le \kappa^{1/2}\int_{-\infty}^t\big(g(t-u)-g(\!-u)\big)^2K_2(u)\,{\mathrm{d}} u, \end{equation}

which is finite by using Conditions 1 and 2, with $\gamma\in(0,1)$ and $\beta\in\big(\frac12-{\gamma}/{2},1\big)$ .

Putting (29)–(33) together, we obtain (28). The proof is complete.

For any $t\ge 0$ and $u\in\mathbb R\setminus\{0\}$ , let $g_{\beta,\gamma}(t,u) \,:\!= \big((t-u)_+^{1-\beta}-(-u)_+^{1-\beta}\big)|u|^{-{\gamma}/{2}}$ . For any $\boldsymbol{t}=(t_1,\ldots,t_m)^\top\in \mathbb R_+^m$ , let $M_{\beta,\gamma}(\boldsymbol{t})\,:\!=\big(M_{\beta,\gamma}(t_i,t_j)\big)_{i,j\in\{1,\ldots,m\}}$ be the $m\times m$ matrix with entries $M_{\beta,\gamma}(t_i,t_j)\,:\!=\int_{\mathbb R}g_{\beta,\gamma}(t_i,u)g_{\beta,\gamma}(t_j,u)\,{\mathrm{d}} u$ .

Lemma 3. Under the conditions of Theorem 2, for any $m\in\mathbb N$ , ${\boldsymbol\theta}=(\theta_1,\ldots,\theta_m)^\top\in\mathbb R^m$ and $\boldsymbol{t}=(t_1,\ldots,t_m)^\top\in\mathbb R_+^m$ , the following limit exists:

(34) \begin{equation} \lim_{n\rightarrow\infty}\frac{1}{\pi(n)^2}\log\mathbb E\Bigg[ \exp\!\Bigg(\pi(n)^2\sum_{i=1}^m\theta_iS_n(t_i)\Bigg)\Bigg] = \frac{1}{2}\left\langle{\boldsymbol\theta},M_{\beta,\gamma}(\boldsymbol{t}){\boldsymbol \theta}\right\rangle. \end{equation}

Consequently, as $n\rightarrow\infty$ , the sequence $\{(S_n(t_1),\ldots,S_n(t_m))\}_{n\in\mathbb N_*}$ satisfies a large deviation principle on $\mathbb R^m$ with speed function $\pi(n)^2$ and rate function $J^{M_{\beta,\gamma}(\boldsymbol{t})}$ defined for ${\boldsymbol{x}}\in\mathbb R^m$ by

$$J^{M_{\beta,\gamma}(\boldsymbol{t})}(\boldsymbol{x}) \,:\!= {\sup}_{\boldsymbol{y}\in\mathbb R^m}\bigg\{ {\boldsymbol{x}}^\top{\boldsymbol{y}}-\frac12{\boldsymbol{y}}^\top M_{\beta,\gamma}(\boldsymbol{t}){\boldsymbol{y}}\bigg\}.$$

Proof. First, let us state the following two facts, which will be used frequently:

1. (i) Under Conditions 2, 3, and 6, it follows from (22) that

   (35) \begin{equation} \lim_{n\rightarrow\infty}{\sup}_{-\infty < u\le t\le T}\frac{\pi(n)}{b(n)}|g(nt -nu)-g(\!-nu)| = 0. \end{equation}

2. (ii) Under Conditions 2, 3, and 4, we derive, as in [Reference Pang and Taqqu35, (52)], that

   (36) \begin{equation} \lim_{n\rightarrow\infty}\frac{n}{b(n)^2}\big(g(nt -nu)-g(\!-nu)\big)^2 K_2(nu) = \big((t-u)_+^{1-\beta}-(\!-u)_+^{1-\beta}\big)^2|u|^{-\gamma}. \end{equation}
   Furthermore, it follows from the dominated convergence theorem, (23), and (36) that
   (37) \begin{multline} \lim_{n\rightarrow\infty}\int_{-\infty}^t\frac{n}{b(n)^2}\big(g(nt-nu)-g(\!-nu)\big)^2K_2(nu)\,{\mathrm{d}} u \\ = \int_{-\infty}^t\big((t-u)_+^{1-\beta}-(\!-u)_+^{1-\beta}\big)^2|u|^{-\gamma}\,{\mathrm{d}} u. \end{multline}

We first prove (34) in the case $m=1$ , that is,

(38) \begin{equation} \lim_{n\rightarrow\infty}\frac{1}{\pi(n)^2}\log\mathbb E[\exp\!(\theta\pi(n)^2S_n(t))] = \frac{\theta^2}{2}\int_{-\infty}^t\big((t-u)_+^{1-\beta}-(\!-u)_+^{1-\beta}\big)^2|u|^{-\gamma}\,{\mathrm{d}} u. \end{equation}

The proof of (38) is divided into two parts: the upper bound and the lower bound.

Step 1: Upper bound. For any fixed $\theta\in\mathbb R$ , $t\ge0$ , by (2), (26), and (35), we have

(39) \begin{align} & \frac{1}{\pi(n)^2}\log\mathbb E[\exp\!(\theta\pi(n)^2S_n(t))] \nonumber \\ & \qquad = \frac{1}{\pi^{2}(n)}\int_{-\infty}^{at}\int_{\mathbb R}\bigg[ \exp\!\bigg(\theta\frac{\pi(n)}{b(n)}\big(g(nt-u)-g(\!-u)\big)r\bigg) - 1\bigg]\,{\mathrm{d}} F_u(r)\,{\mathrm{d}} u \nonumber \\ & \qquad = \frac{1}{\pi^{2}(n)}\int_{-\infty}^t\int_{\mathbb R}\bigg[ \exp\!\bigg(\theta\frac{\pi(n)}{b(n)}\big(g(nt-nu)-g(\!-nu)\big)r\bigg) - 1 \nonumber \\ & \qquad\qquad\qquad\qquad\qquad - \theta\frac{\pi(n)}{b(n)}\big(g(nt-nu)-g(\!-nu)\big)r\bigg]\,{\mathrm{d}} F_{nu}(r)\,n{\mathrm{d}} u. \end{align}

For any $\varepsilon>0$ , there exists a constant $c_{4,1}(\varepsilon)>0$ such that

(40) \begin{equation} \bigg|{\mathrm{e}}^{x}-1-x-\frac{x^2}{2}\bigg| \le \frac{\varepsilon}{2}x^2 \quad \text{for all } |x| \le c_{4,1}(\varepsilon). \end{equation}

It follows from (35) and (40) that, for any $\varepsilon>0$ and $r_0>0$ ,

(41) \begin{align} & \limsup_{n\rightarrow\infty}\frac{1}{\pi^{2}(n)}\bigg(\int_{-\infty}^t\int_{|r|\le r_0}\bigg[ \exp\!\bigg(\theta\frac{\pi(n)}{b(n)}\big(g(nt-nu)-g(\!-nu)\big)r\bigg) - 1 \nonumber \\ & \qquad\qquad\qquad\qquad\qquad\qquad - \theta\frac{\pi(n)}{b(n)}\big(g(nt-nu)-g(\!-nu)\big)r\bigg]\,{\mathrm{d}} F_{nu}(r)\,n{\mathrm{d}} u\bigg) \nonumber \\ & \qquad \le \frac{(1+\varepsilon)\theta^2}{2}\limsup_{n\rightarrow\infty}\int_{-\infty}^t\int_{|r|\le r_0}\frac{n}{b(n)^2} \big(g(nt-nu)-g(\!-nu)\big)^2r^2\,{\mathrm{d}} F_{nu}(r)\,{\mathrm{d}} u \nonumber \\ & \qquad \le \frac{(1+\varepsilon)\theta^2}{2}\lim_{n\rightarrow\infty}\int_{-\infty}^t\frac{n}{b(n)^2} \big(g(nt-nu)-g(\!-nu)\big)^2K_2(nu)\,{\mathrm{d}} u \nonumber \\ & \qquad = \frac{(1+\varepsilon)\theta^2}{2}\int_{-\infty}^t\big((t-u)_+^{1-\beta}-(\!-u)_+^{1-\beta}\big)^2 |u|^{-\gamma}\,{\mathrm{d}} u, \end{align}

where (37) was used in the last step.

It follows from (30) and the Cauchy–Schwarz inequality that

(42) \begin{align} & \int_{|r|>r_0}\!\bigg[\exp\!\bigg(\theta\frac{\pi(n)}{b(n)}\big(g(nt-nu)-g(\!-nu)\big)r\bigg) - 1 - \theta\frac{\pi(n)}{b(n)}\big(g(nt-nu)-g(\!-nu)\big)r\bigg]\,{\mathrm{d}} F_{nu}(r) \nonumber \\ & \le \theta^2\frac{\pi(n)^2}{b(n)^2}\big(g(nt-nu)-g(\!-nu)\big)^2\!\int_{|r|>r_0}|r|^2 \exp\!\bigg(\theta\frac{\pi(n)}{b(n)}|\big(g(nt-nu)-g(\!-nu)\big)r|\bigg)\,{\mathrm{d}} F_{nu}(r) \nonumber \\ & \le \theta^2\frac{\pi(n)^2}{b(n)^2}\big(g(nt-nu)-g(\!-nu)\big)^2 \bigg(\int_{\mathbb R}|r|^4\,{\mathrm{d}} F_{nu}(r)\bigg)^{{1}/{2}} \nonumber \\ & \quad \times \bigg(\int_{|r|>r_0}\exp\!\bigg(2\theta\frac{\pi(n)}{b(n)}|\big(g(nt-nu)-g(\!-nu)\big)r|\bigg)\, {\mathrm{d}} F_{nu}(r)\bigg)^{{1}/{2}}. \end{align}

By (35) and Chebyshev’s inequality we have, for sufficiently large n,

(43) \begin{align} & \int_{|r|>r_0}\exp\!\bigg(2\theta\frac{\pi(n)}{b(n)}|\big(g(nt-nu)-g(\!-nu)\big)r|\bigg)\,{\mathrm{d}} F_{nu}(r)\nonumber\\ & \qquad \le \int_{|r|>r_0}\exp\!\bigg(\frac{\theta_0}{2}|r|\bigg)\,{\mathrm{d}} F_{nu}(r) \nonumber \\ & \qquad \le \exp\!\bigg({-}\frac{\theta_0r_0}{2}\bigg)\int_{|r|>r_0}\exp\!(\theta_0|r|)\,{\mathrm{d}} F_{nu}(r) \le \exp\!\bigg({-}\frac{\theta_0r_0}{2}\bigg)M_{\theta_0}, \end{align}

where (8) in Condition 4 was used in the last step.

By using Condition 5 and (37), we derive

(44) \begin{align} & \limsup_{n\rightarrow\infty}\frac{n}{b(n)^2}\int_{-\infty}^t\big(g(nt-nu)-g(\!-nu)\big)^2 \bigg(\int_{\mathbb R}|r|^4\,{\mathrm{d}} F_{nu}(r)\bigg)^{{1}/{2}}\,{\mathrm{d}} u \nonumber \\ & \qquad = \limsup_{n\rightarrow\infty}\frac{n}{b(n)^2}\int_{-\infty}^t\big(g(nt-nu)-g(\!-nu)\big)^2 K_4(nu)^{{1}/{2}}\,{\mathrm{d}} u \nonumber \\ & \qquad \le \kappa^{1/2}\limsup_{n\rightarrow\infty}\frac{n}{b(n)^2}\int_{-\infty}^t\big(g(nt-nu)-g(\!-nu)\big)^2 K_2(nu)\,{\mathrm{d}} u \nonumber \\ & \qquad = \kappa^{1/2}\int_{-\infty}^t\big((t-u)_{+}^{1-\beta}-(\!-u)_+^{1-\beta}\big)^2|u|^{-\gamma}\,{\mathrm{d}} u. \end{align}

Putting (39), (41), (42), (43), and (44) together, and using the arbitrariness of $r_0$ , we obtain that, for any $\theta\in\mathbb R$ , $t\ge0$ , and $\varepsilon\in(0,1)$ ,

(45) \begin{multline} \limsup_{n\rightarrow\infty}\frac{1}{\pi(n)^2}\log\mathbb E[\!\exp\!(\theta\pi(n)^2S_n(t))] \\ \le \frac{\theta^2(1+\varepsilon)}{2}\int_{-\infty}^t\big((t-u)_+^{1-\beta}-(\!-u)_+^{1-\beta}\big)^2|u|^{-\gamma}\,{\mathrm{d}} u. \end{multline}

Step 2: Lower bound. Since ${\mathrm{e}}^{x}-1-x \geq 0$ for all $x\in\mathbb R$ , it follows from (35) and (40) that, for any $\varepsilon\in (0,1)$ and $r_0>0$ ,

(46) \begin{align} & \liminf_{n\rightarrow\infty}\frac{1}{\pi^{2}(n)}\bigg(\int_{-\infty}^t\int_{\mathbb R}\bigg[ \exp\!\bigg(\theta\frac{\pi(n)}{b(n)}\big(g(nt-nu)-g(\!-nu)\big)r\bigg) - 1 \nonumber \\ & \qquad\qquad\qquad\qquad\qquad -\theta\frac{\pi(n)}{b(n)}\big(g(nt-nu)-g(\!-nu)\big)r\bigg]\,{\mathrm{d}} F_{nu}(r)\,n{\mathrm{d}} u\bigg) \nonumber \\ & \geq \liminf_{n\rightarrow\infty}\frac{1}{\pi^{2}(n)}\bigg(\int_{-\infty}^t\int_{|r|\le r_0}\bigg[ \exp\!\bigg(\theta\frac{\pi(n)}{b(n)}\big(g(nt-nu)-g(\!-nu)\big)r\bigg) - 1 \nonumber \\ & \qquad\qquad\qquad\qquad\qquad\qquad\quad -\theta\frac{\pi(n)}{b(n)}\big(g(nt-nu)-g(\!-nu)\big)r\bigg]\,{\mathrm{d}} F_{nu}(r)\,n{\mathrm{d}} u\bigg) \nonumber \\ & \geq \frac{(1-\varepsilon)\theta^2}{2}\liminf_{n\rightarrow\infty}\int_{-\infty}^t\int_{|r|\le r_0}\frac{n}{b(n)^2} \big(g(nt-nu)-g(\!-nu)\big)^2r^2\,{\mathrm{d}} F_{nu}(r)\,{\mathrm{d}} u \nonumber \\ & \geq \frac{(1-\varepsilon)\theta^2}{2}\liminf_{n\rightarrow\infty}\int_{-\infty}^t\int_{\mathbb R}\frac{n}{b(n)^2} \big(g(nt-nu)-g(\!-nu)\big)^2K_2(nu)\,{\mathrm{d}} u \nonumber \\ & \quad - \frac{(1-\varepsilon)\theta^2}{2}\limsup_{n\rightarrow\infty}\int_{-\infty}^t\int_{|r|>r_0}\frac{n}{b(n)^2} \big(g(nt-nu)-g(\!-nu)\big)^2r^2\,{\mathrm{d}} F_{nu}(r)\,{\mathrm{d}} u \nonumber \\ & = \frac{(1-\varepsilon)\theta^2}{2}\int_{-\infty}^t\big((t-u)_+^{1-\beta}-(\!-u)_+^{1-\beta}\big)^2 |u|^{-\gamma}\,{\mathrm{d}} u \nonumber \\ & \quad - \frac{(1-\varepsilon)\theta^2}{2}\limsup_{n\rightarrow\infty}\int_{-\infty}^t\int_{|r|>r_0}\frac{n}{b(n)^2} \big(g(nt-nu)-g(\!-nu)\big)^2r^2\,{\mathrm{d}} F_{nu}(r)\,{\mathrm{d}} u, \end{align}

where (37) was used in the last step. Next, we prove that the last term in (46) can be ignored as $r_0\to \infty$ .

By using the Cauchy–Schwarz inequality and Condition 5, we have

(47) \begin{align} & \int_{-\infty}^t\int_{|r|>r_0}\frac{n}{b(n)^2}\big(g(nt-nu)-g(\!-nu)\big)^2r^2\,{\mathrm{d}} F_{nu}(r)\,{\mathrm{d}} u \nonumber \\ & \quad \le \frac{n}{b(n)^2}\int_{-\infty}^t\big(g(nt-nu)-g(\!-nu)\big)^2 \bigg(\int_{\mathbb R}|r|^4\,{\mathrm{d}} F_{nu}(r)\bigg)^{{1}/{2}} \bigg(\int_{|r|>r_0}\,{\mathrm{d}} F_{nu}(r)\bigg)^{{1}/{2}}\,{\mathrm{d}} u \nonumber \\ & \quad \le \frac{\kappa^{1/2}n}{b(n)^2}\int_{-\infty}^t\big(g(nt-nu)-g(\!-nu)\big)^2K_2(nu)\,{\mathrm{d}} u \cdot {\sup}_{u\in\mathbb R}\bigg(\int_{|r|>r_0}\,{\mathrm{d}} F_{nu}(r)\bigg)^{{1}/{2}}. \end{align}

By Chebyshev’s inequality and Condition 4, we have

(48) \begin{align} {\sup}_{u\in\mathbb R}\int_{|r|>r_0}{\mathrm{d}} F_{nu}(r) & \le \exp\!(\!-\theta_0r_0) \cdot {\sup}_{u\in\mathbb R}\int_{|r|>r_0}\exp\!(\theta_0|r|)\,{\mathrm{d}} F_{nu}(r) \nonumber \\ & \le \exp\!(\!-\theta_0r_0)M_{\theta_0}. \end{align}

Putting (39), (46), (47), and (48) together, and using the arbitrariness of $r_0$ , we obtain that, for any $\theta\in\mathbb R$ , $t\ge0$ , and $\varepsilon\in(0,1)$ ,

\begin{equation*} \liminf_{n\rightarrow\infty}\frac{1}{\pi(n)^2}\log\mathbb E[\exp\!(\theta\pi(n)^2S_n(t))] \ge \frac{\theta^2(1-\varepsilon)}{2}\int_{-\infty}^t\big((t-u)_+^{1-\beta}-(\!-u)_+^{1-\beta}\big)^2|u|^{-\gamma}\,{\mathrm{d}} u. \end{equation*}

This, together with (45), implies (38).

Next, we prove (34) for $m\ge2$ . Notice that, for any ${\boldsymbol{\theta}}\,:\!=(\theta_1,\ldots,\theta_m)^\top\in\mathbb R^m$ and $\boldsymbol{t}\,:\!=(t_1,\ldots,t_m)^\top\in\mathbb R_+^m$ , we can write

\begin{multline*} \mathbb E\Bigg[\exp\!\Bigg(\frac{\pi(n)}{b(n)}\sum_{i=1}^m\theta_iS(nt_i)\Bigg)\Bigg] \\ = \mathbb E\Bigg[\exp\!\Bigg(\frac{\pi(n)}{b(n)}\int_{\mathbb R}\int_{\mathbb R} \Bigg(\sum_{j=1}^k\theta_j\big(g(nt_j-u)-g(\!-u)\big)\Bigg)r\,N({\mathrm{d}} u,{\mathrm{d}} r)\Bigg)\Bigg]. \end{multline*}

The proof of (34) is similar to that of (38). More specifically, we replace $\theta\big(g(nt-u)-g(\!-u)\big)$ in (39) and (46) with $\sum_{j=1}^m\theta_j\big(g(nt_j-u)-g(\!-u)\big)$ and follow the same computation to obtain

\begin{align*} & \lim_{n\rightarrow\infty}\frac{1}{\pi(n)^2}\log\mathbb E\Bigg[ \exp\!\Bigg(\pi(n)^2\sum_{i=1}^m\theta_iS_n(t_i)\Bigg)\Bigg] \\ & \qquad = \frac12\int_{-\infty}^{\infty}\Bigg(\sum_{j=1}^m\theta_j\big((t_j-u)_+^{1-\beta}-(\!-u)_+^{1-\beta}\big)\Bigg)^2 |u|^{-\gamma}\,{\mathrm{d}} u \\ & \qquad = \frac12\big\langle{\boldsymbol{\theta}},M_{\beta,\gamma}({\boldsymbol{t}}){\boldsymbol{\theta}}\big\rangle. \end{align*}

Thus, (34) holds. By the Gärtner–Ellis theorem (see, e.g., [Reference Dembo and Zeitouni12, Theorem 2.3.6]), we obtain the desired result.

4.2. MDP in the topology of pointwise convergence

Proof. It follows from the Dawson–Gärtner theorem [Reference Dembo and Zeitouni12, Theorem 4.6.1] and Lemma 3 that the family $\{S_n(t)\colon t\in[0,T]\}_{n\in\mathbb N_*}$ satisfies a large deviation principle on $\mathcal C[0,T]$ in the topology of pointwise convergence as $n\rightarrow\infty$ , with the speed function $\pi(n)^2$ and the rate function $I_{\beta,\gamma}'$ given by

(49) \begin{equation} I_{\beta,\gamma}'(\xi) \,:\!= {\sup}_{0\le t_1 < \cdots < t_m\le T,m\in\mathbb N}J^{M_{\beta,\gamma}(t_1,\ldots,t_m)} ((\xi(t_1),\ldots,\xi_{t_m})^\top). \end{equation}

Following exactly the same steps as in the proof of [Reference Majewski33, Lemma 11], we conclude that the rate functions $I'_{\beta,\gamma}$ in (49) and $I_{\beta,\gamma}$ in (11) agree on $\mathcal C[0,T]$ . The proof is complete.

4.3. Exponential tightness in $\big(\mathcal C[0, T], \|\cdot\|_T\big)$

To strengthen the MDP to the topology of uniform convergence, we establish the associated exponential tightness of $\{S_n(t)\colon t\in[0,T]\}_{n\in\mathbb N_*}$ .

Lemma 6. Under the conditions of Theorem 2, for any $T>0$ there exist some positive constants $c_{4,2}$ , $c_{4,3}$ , and $c_{4,4}$ such that, for any $n\ge c_{4,2}$ , and $0\le s$ , $t\le T$ , with $0 < \eta < \frac{3}{2}-\beta-{\gamma}/{2}$ , the following inequality holds:

(50) \begin{equation} \mathbb E\bigg[\exp\!\bigg(\pi(n)^2\frac{|S_n(t)-S_n(s)|}{|t-s|^{\eta}}\bigg)\bigg] \le c_{4,3}\exp\!(c_{4,4}\pi(n)^2). \end{equation}

Proof. By using the same argument as in the proof of (38) we obtain, for any $\theta\in\mathbb R$ and $0\le s< t\le T$ ,

(51) \begin{align} \lim_{n\rightarrow\infty}\frac{1}{\pi(n)^2}\log\mathbb E[\exp\!(\theta\pi(n)^2(S_n(t)-S_n(s))] & = \frac{\theta^2}{2}\int_{-\infty}^t \big((t-u)_+^{1-\beta}-(s-u)_+^{1-\beta}\big)^2|u|^{-\gamma}\,{\mathrm{d}} u \nonumber \\ & \le c_{2,1}(\gamma,\beta)\frac{\theta^2}{2}|t-s|^{3-2\beta-\gamma}, \end{align}

where (7) was used in the last step. Taking $\theta=\pm{1}/{|t-s|^{\eta}}$ in (51), we have

\begin{align*} & \limsup_{n\rightarrow\infty}\frac{1}{\pi(n)^2}\log\mathbb E\bigg[ \exp\!\bigg(\pi(n)^2\frac{|S_n(t)-S_n(s)|}{|t-s|^{\eta}}\bigg)\bigg] \\ & \qquad = \max\bigg\{\lim_{n\rightarrow\infty}\frac{1}{\pi(n)^2}\log\mathbb E\bigg[ \exp\!\bigg(\pi(n)^2\frac{S_n(t)-S_n(s)}{|t-s|^{\eta}}\bigg)\bigg], \\ & \qquad\qquad\quad\ \ \lim_{n\rightarrow\infty}\frac{1}{\pi(n)^2}\log\mathbb E\bigg[ \exp\!\bigg({-}\pi(n)^2\frac{S_n(t)-S_n(s)}{|t-s|^{\eta}}\bigg)\bigg]\bigg\} \\ & \qquad \le c_{2,1}(\gamma,\beta)|t-s|^{3-2\beta-\gamma-2\eta}. \end{align*}

This implies (50). The proof is complete.