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

TL;DR

This paper establishes weak convergence results for renewal shot noise processes with increasing response functions, revealing their limits as stable Lévy processes, inverse stable subordinators, or fractionally integrated variants, depending on the response function's properties.

Contribution

It provides the first comprehensive weak convergence framework for renewal shot noise processes with regularly varying response functions, including new limit process characterizations.

Findings

Convergence to spectrally nonpositive stable Lévy processes, including Brownian motion.

Limit processes include inverse stable subordinators and their fractional integrations.

Results depend on the regular variation index of the response function.

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 $D[0,\infty)$ under the $J_1$ or $M_1$ topology. The limiting processes are either spectrally nonpositive stable L\'{e}vy 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 L\'{e}vy processes and the continuous mapping theorem.