Modern Stochastics: Theory and Applications

Let $(X_{k},\xi _{k})_{k\in \mathbb{N}}$ be a sequence of independent copies of a pair $(X,\xi )$ where X is a random process with paths in $D[0,\infty )$ and ξ is a positive random variable. We impose no conditions on the dependence structure of $(X,\xi )$. Hereafter $\mathbb{N}_{0}$ denotes the set of non-negative integers $\{0,1,2,\dots \}$.

Let $(S_{n})_{n\in \mathbb{N}_{0}}$ be a standard zero-delayed random walk: and let $(\nu (t))_{t\in \mathbb{R}}$ be the corresponding first-passage time process for $(S_{n})_{n\in \mathbb{N}_{0}}$: The random process with immigration $Y=(Y(u))_{u\in \mathbb{R}}$ is defined as a finite sum This family of random processes was introduced in [11] as a generalization of several known objects in applied probability including branching processes with immigration (in case of X being a branching process) and renewal shot noise processes (in case of $X(t)=h(t)$ a.s. for some $h\in D[0,\infty )$). The process X is usually called a response process, or a response function if $X(t)=h(t)$ a.s. for some deterministic function h.

The problem of weak convergence of random processes with immigration was addressed in [11, 12, 16] where the authors give a more or less complete picture of the weak convergence of finite-dimensional distributions of $(Y(ut))_{u\ge 0}$ or $(Y(u+t))_{u\in \mathbb{R}}$, as $t\to \infty $. The case of renewal shot noise process has received much attention in the past years, see [6, 9, 10, 14]. A comprehensive survey of the subject is given in Chapter 3 of the recent book [7].

A much more delicate question of weak convergence of Y in functional spaces, to the best of our knowledge, was only investigated either for particular response processes, or in the simple case when ξ is exponentially distributed. In the latter situation Y is called a Poisson shot noise process. In the list below η is a random variable which satisfies certain assumptions specified in the corresponding papers:

In this paper we treat the case where ξ is heavy-tailed, more precisely we assume that for some $\ell _{\xi }$ slowly varying at infinity, and $\alpha \in (0,1)$. Assuming (2), we obtain a functional limit theorem for a quite general class of response processes. The class of such processes can be described by a common property: they do not “oscillate to much” around the mean $\mathbb{E}[X(t)]$, which itself varies regularly with parameter $\rho >-\alpha $. Let us briefly outline our approach based on ideas borrowed from [11]. Put $h(t):=\mathbb{E}[X(t)]$ and write1 We investigate the two summands in the right-hand side separately. The second summand is a standard renewal shot noise process with response function h. Under condition (2) and assuming that for some $\rho \in \mathbb{R}$ and a slowly varying function $\ell _{h}$, it was proved in [10, Theorem 2.9] and [14, Theorem 2.1] that where $J_{\alpha ,\rho }=(J_{\alpha ,\rho }(u))_{u\ge 0}$ is a so-called fractionally integrated inverse α-stable subordinator. The process $J_{\alpha ,\rho }$ is defined as a pathwise Lebesgue–Stieltjes integral In this formula ${W_{\alpha }^{\gets }}(y):=\inf \{t\ge 0:W_{\alpha }(t)>y\}$, $y\ge 0$, is a generalized inverse of an α-stable subordinator $(W_{\alpha }(t))_{t\ge 0}$ with the Laplace exponent It is also known that convergence of finite-dimensional distributions (5) can be strengthened to convergence in the Skorokhod space $D(0,\infty )$ endowed with the $J_{1}$-topology if $\rho >-\alpha $, see Theorem 2.1 in [14]. If $\rho \le -\alpha $ the process $(J_{\alpha ,\rho }(u))_{u\ge 0}$, being a.s. finite for every fixed $u\ge 0$, has a.s. locally unbounded trajectories, see Proposition 2.5 in [14].

Turning to the first summand in (3) we note that it is the a.s. limit of a martingale $(R(j,t),\mathcal{F}_{j})_{j\in \mathbb{N}}$, where $\mathcal{F}_{j}:=\sigma ((X_{k},\xi _{k}):1\le k\le j)$ and Applying the martingale central limit theory it is possible to show that under appropriate assumptions (which are of no importance for this paper) as $t\to \infty $, for a non-trivial process Z, where $v(t):=\mathbb{E}[{(X(t)-h(t))}^{2}]$ is the variance of X, see Proposition 2.2 in [11].

We are interested in situations when the second summand in (3) asymptotically dominates, more precisely we are looking for conditions ensuring for every fixed $T>0$. From what has been mentioned above it is clear that this can happen only if Restricting our attention to the case where v is regularly varying with index $\beta \in \mathbb{R}$, i.e. we see that (8) holds if $\beta <\alpha +2\rho $ and fails if $\beta >\alpha +2\rho $. As long as we do not make any assumptions on distributional or path-wise properties of X such as e.g., monotonicity, self-similarity or independence of increments, it can be hardly expected that condition (8) alone is sufficient for (7). Nevertheless, we will show that (7) holds true under additional assumptions on the asymptotic behavior of higher centered moments $\mathbb{E}[{(X(t)-h(t))}^{2l}]$, $l=1,2,\dots $, and an additional technical assumption. Our first main result treats the case where the moments of the normalized process $([X(t)-h(t)]/v(t))_{t\ge 0}$ are bounded uniformly in $t\ge 0$. Denote by $(\widehat{X}(t))_{t\ge 0}$ the centered process $(X(t)-h(t))_{t\ge 0}$.

Our second main result is mainly applicable when the process X is almost surely bounded by some (deterministic) constant. We have the following theorem.

Obviously, our results are far from being optimal and leave a lot of space for improvements, yet they are applicable to several models given in the next section.

The proofs of Theorems 1 and 2 rely on the same ideas, so we will prove them simultaneously. Pick $\delta >0$ such that all assumptions of Theorem 1 or Theorem 2 hold. This $\delta >0$ remains fixed throughout the proof.

In view of assumptions (A1) and (A2) and the fact that h is càdlàg we infer from Theorem 2.1 in [14] that weakly on $D(0,\infty )$ endowed with the $J_{1}$-topology. Note that in Theorem 2.1 of [14] h is assumed monotone (or eventually monotone). However, this assumption is redundant. The only places which have to be adjusted in the proofs are two displays on p. 90, where $h(0)$ should be replaced by $\sup _{y\in [0,c]}h(y)$.

Hence, from (3) we see that it is enough to check, for every fixed $T>0$, that where $\widetilde{Y}(t):=\sum _{k\ge 0}(X_{k+1}(t-S_{k})-h(t-S_{k}))\mathbb{1}_{\{S_{k}\le t\}}$ for $t\ge 0$. Moreover, it suffices to show that Indeed, for every fixed $s>0$, Since $t\mapsto h(t)/\mathbb{P}\{\xi >t\}$ is regularly varying with positive index $\rho +\alpha $, Sending $t\to \infty $ we obtain, for every fixed $s>0$, Sending now $s\to \infty $ shows that (19) implies (18). Let us first check that (19) holds along the arithmetic sequence $(n\delta )_{n\in \mathbb{N}}$. According to the Borel–Cantelli lemma and Markov’s inequality it suffices to check that for some $l\in \mathbb{N}$ To check (20) we apply the Burkholder–Davis–Gundy inequality in the form given in Theorem 11.3.2 of [4], to obtain for some constant $K_{l}>0$, where we recall the notation $\mathcal{F}_{k}=\sigma ((X_{j},\xi _{j}):1\le j\le k)$.