Weak convergence of renewal shot noise processes in the case of slowly varying normalization

TL;DR

This paper studies the weak convergence of renewal shot noise processes with slowly varying normalization, revealing that fluctuations are approximated by a combination of Brownian motion and a Gaussian process under specific conditions.

Contribution

It extends the understanding of shot noise processes by analyzing the case where the response function is regularly varying with index -1/2 and the integral of its square diverges, using strong approximation techniques.

Findings

Fluctuations occur on a scale depending on the slowly varying function.

Finite-dimensional distributions are approximated by a sum of independent Brownian motion and Gaussian process.

Results apply when the response function's scaled limit is finite and positive.

Abstract

We investigate weak convergence of finite-dimensional distributions of a renewal shot noise process $(Y(t))_{t\geq 0}$ with deterministic response function $h$ and the shots occurring at the times $0 = S_0 < S_1 < S_2<\ldots$, where $(S_n)$ is a random walk with i.i.d.\ jumps. There has been an outbreak of recent activity around this topic. We are interested in one out of few cases which remained open: $h$ is regularly varying at $\infty$ of index $-1/2$ and the integral of $h^2$ is infinite. Assuming that $S_1$ has a moment of order $r>2$ we use a strong approximation argument to show that the random fluctuations of $Y(s)$ occur on the scale $s=t+g(t,u)$ for $u\in [0,1]$, as $t\to\infty$, and, on the level of finite-dimensional distributions, are well approximated by the sum of a Brownian motion and a Gaussian process with independent values (the two processes being independent). The scaling function $g$ above depends on the slowly varying factor of $h$. If, for instance, $\lim_{t\to\infty}t^{1/2}h(t)\in (0,\infty)$, then $g(t,u)=t^u$.