2015/07/09 by Alexander Iksanov, Iksanov, Alexander, Zakhar Kabluchko +3
Computer Science · Decision Sciences · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.1507.02526
17 pages, to appear in Statistics and Probability Letters
openalex publication_date 2015/07/09 · arxiv created 2016/03/13 · arxiv updated 2016/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate weak convergence of finite-dimensional distributions of a renewal shot noise process (Y(t))t≥ 0 with deterministic response function h and the shots occurring at the times 0 = S0 < S1 < S2<…, where (Sn) 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 ∞ of index -1/2 and the integral of h2 is infinite. Assuming that S1 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∈ [0,1], as t→∞, 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, limt→∞t1/2h(t)∈ (0,∞), then g(t,u)=tu.