Functional limit theorems for renewal shot noise processes with increasing response functions

Alexander Iksanov

Stochastic Processes and their Applications, 2013, vol. 123, issue 6, 1987-2010

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,∞) 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.

Keywords: Continuous mapping theorem, fractionally integrated (inverse) stable process; Functional limit theorem; M1 topology; Renewal shot noise process; Spectrally negative stable process (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (6)

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DOI: 10.1016/j.spa.2013.01.019

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