Limit theorems for renewal shot noise processes with decreasing response functions

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 $t\to \mathrm{\infty }$ 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 $t\to \mathrm{\infty }$. 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 $\mathrm{\infty }$ with index $\beta$ (with $\beta <1\mathrm{/}\alpha$ or $\beta \leq 1/\alpha$,dependingonwhether$$\mathbb{E}$ S_1 < \infty$or$$\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 L'{e}vy motions if $$\mathbb{E}$ S_1 < \infty$andfractionallyintegratedinversestablesubordinatorsif$$\mathbb{E}$ S_1 = \infty$.