Limit theorems for renewal shot noise processes with decreasing response functions

TL;DR

This paper establishes limit theorems for renewal shot noise processes with decreasing response functions, identifying conditions for weak convergence and describing the limiting processes in various regimes.

Contribution

It provides new limit theorems for renewal shot noise processes with decreasing response functions, including stable and inverse stable process limits.

Findings

Weak convergence of finite-dimensional distributions under integrable response functions.

Convergence after shifting for non-integrable decreasing response functions.

Identification of limiting processes as fractionally integrated stable Lévy motions or inverse stable subordinators.

Abstract

We consider shot noise processes $(X(t))_{t \geq 0}$ with deterministic response function $h$ and the shots occurring at the renewal epochs $0= S_0 < S_1 < S_2 ...$ of a zero-delayed renewal process. We prove convergence of the finite-dimensional distributions of $(X(ut))_{u \geq 0}$ as $t \to \infty$ in different regimes. If the response function $h$ is directly Riemann integrable, then the finite-dimensional distributions of $(X(ut))_{u \geq 0}$ converge weakly as $t \to \infty$. Neither scaling nor centering are needed in this case. If the response function is eventually decreasing, non-integrable with an integrable power, then, after suitable shifting, the finite-dimensional distributions of the process converge. Again, no scaling is needed. In both cases, the limit is identified. If the distribution of $S_1$ is in the domain of attraction of an $\alpha$-stable law and the response function is regularly varying at $\infty$ with index $\beta$ (with $\beta < 1/\alpha$ or $\beta \leq 1/\alpha$, depending on whether $\mathbb{E} S_1 < \infty$ or $\mathbb{E} S_1 = \infty$), then scaling is needed to obtain weak convergence of the finite-dimensional distributions of $(X(ut))_{u \geq 0}$. The limiting processes are fractionally integrated stable L\'{e}vy motions if $\mathbb{E} S_1 < \infty$ and fractionally integrated inverse stable subordinators if $\mathbb{E} S_1 = \infty$.