Limit theorems for renewal shot noise processes with decreasing response functions

Abstract

We consider shot noise processes $(X(t){)}_{t\ge 0}$ 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 $(X(ut){)}_{u\ge 0}$ as $t\to \infty$ in different regimes. If the response function $h$ is directly Riemann integrable, then the finite-dimensional distributions of $(X(ut){)}_{u\ge 0}$ 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 $\beta \le 1/\alpha$, 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 $(X(ut){)}_{u\ge 0}$. The limiting processes are fractionally integrated stable Lévy motions if $\mathbb{E}{S}_1<\infty$ and fractionally integrated inverse stable subordinators if $\mathbb{E}{S}_1=\infty$.

58 pages, submitted in a shortened form; the present version corrects a coupling defined in Section 3.1 and used in various parts of the paper