Article ID Journal Published Year Pages File Type
1156746 Stochastic Processes and their Applications 2010 24 Pages PDF
Abstract

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,∞)D[0,∞) under the J1J1 or M1M1 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.

Related Topics
Physical Sciences and Engineering Mathematics Mathematics (General)
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