Weak convergence of renewal shot noise processes in the case of slowly varying normalization

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• 作者：Alexander Iksanov^a ; ^{iksan@univ.kiev.ua" class="auth_mail" title="E-mail the corresponding author} ; Zakhar Kabluchko^b ; ^{zakhar.kabluchko@uni-muenster.de" class="auth_mail" title="E-mail the corresponding author} ; Alexander Marynych^a ; ^b ; ^{marynych@unicyb.kiev.ua" class="auth_mail" title="E-mail the corresponding author}
• 关键词：60F05 ; 60K05
• 刊名：Statistics & Probability Letters
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：114
• 期：Complete
• 页码：67-77
• 全文大小：592 K

文摘

We investigate weak convergence of finite-dimensional distributions of a renewal shot noise process (Y(t))_t≥0${\left(Y\left(t\right)\right)}_{t\ge 0}$ with deterministic response function h$h$ and the shots occurring at the times 0=S₀<S₁<S₂<…$0=S_0<S_1<S_2<\dots$, where (S_n)$\left(S_n\right)$ 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$h$ is regularly varying at ∞$\infty$ of index −1/2$-1/2$ and the integral of h²$h^2$ is infinite. Assuming that S₁$S_1$ has a moment of order r>2$r>2$ we use a strong approximation argument to show that the random fluctuations of Y(s)$Y\left(s\right)$ occur on the scale 2255ac3c37c052191f5cf6" title="Click to view the MathML source">s=t+g(t,u)$s=t+g(t,u)$ for u∈[0,1]$u\in [0,1]$, as t→∞$t\to \infty$, 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$g$ above depends on the slowly varying factor of h$h$. If, for instance, lim_t→∞t^1/2h(t)∈(0,∞)${lim}_{t\to \infty }{t}^{1/2}h\left(t\right)\in (0,\infty )$, then g(t,u)=t^u$g(t,u)=t^u$.