2012/12/07 by A. Iksanov, Alexander Iksanov, A. Marynych +6
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #math.PR
paper · pdf · doi:10.48550/arxiv.1212.1583
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
openalex publication_date 2012/12/07 · arxiv created 2013/10/24 · arxiv updated 2013/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider shot noise processes (X(t))t ≥ 0 with deterministic response function h and the shots occurring at the renewal epochs 0= S0 < S1 < S2 ... of a zero-delayed renewal process. We prove convergence of the finite-dimensional distributions of (X(ut))u ≥ 0 as t → ∞ in different regimes. If the response function h is directly Riemann integrable, then the finite-dimensional distributions of (X(ut))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 S1 is in the domain of attraction of an α-stable law and the response function is regularly varying at ∞ with index β (with β< 1/α or β≤ 1/α, depending on whether 𝔼 S1 < ∞ or 𝔼 S1 = ∞), then scaling is needed to obtain weak convergence of the finite-dimensional distributions of (X(ut))u ≥ 0. The limiting processes are fractionally integrated stable Lévy motions if 𝔼 S1 < ∞ and fractionally integrated inverse stable subordinators if 𝔼 S1 = ∞.