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

Abstract

We investigate weak convergence of finite-dimensional distributions of a renewal shot noise process $(Y(t){)}_{t\ge 0}$ with deterministic response function $h$ and the shots occurring at the times $0=S_0<S_1<S_2<\dots$, 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 $g(t,u)=t^u$.

arXiv comment17 pages, to appear in Statistics and Probability Letters