2012/02/09 by Alexander Iksanov, Iksanov, Alexander · 1 citation
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.1202.1950
Stoch. Proc. Appl., accepted for publication
openalex publication_date 2012/02/09 · arxiv created 2013/01/29 · arxiv updated 2013/01/30 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We consider renewal shot noise processes with response functions which are eventually nondecreasing and regularly varying at infinity. We prove weak convergence of renewal shot noise processes, properly normalized and centered, in the space D[0,∞) under the J1 or M1 topology. The limiting processes are either spectrally nonpositive stable Lévy processes, including the Brownian motion, or inverse stable subordinators (when the response function is slowly varying), or fractionally integrated stable processes or fractionally integrated inverse stable subordinators (when the index of regular variation is positive). The proof exploits fine properties of renewal processes, distributional properties of stable Lévy processes and the continuous mapping theorem.