Limit theorems for renewal shot noise processes with eventually decreasing response functions

We consider shot noise processes ( X ( t ) ) t ≥ 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 ( u t ) ) u ≥ 0 as t → ∞ in different regimes. If the response function h is directly Riemann integrable, then the finite-dimensional distributions of ( X ( u t ) ) u ≥ 0 converge weakly as t → ∞ . 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 α -stable law and the response function is regularly varying at ∞ with index − β (with 0 ≤ β < 1 / α or 0 ≤ β ≤ α , depending on whether E S 1 < ∞ or E S 1 = ∞ ), then scaling is needed to obtain weak convergence of the finite-dimensional distributions of ( X ( u t ) ) u ≥ 0 . The limits are fractionally integrated stable Lévy motions if E S 1 < ∞ and fractionally integrated inverse stable subordinators if E S 1 = ∞ .