# A functional limit theorem for general shot noise processes

## TL;DR

This paper establishes a general functional limit theorem for shot noise processes with arbitrary shot counting, showing convergence to different Gaussian processes depending on the limit of the counting process, and explores the regularity of these limits.

## Contribution

It extends existing limit theorems to more general shot noise processes with arbitrary counting, identifying new limit processes and normalization methods.

## Key findings

- Limit processes include Riemann-Liouville processes when the counting process converges to Brownian motion.
- The results apply to five specific counting processes.
- The paper investigates the Hölder continuity of the limit processes.

## Abstract

By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally H\"{o}lder continuous Gaussian limit process and that the response function is regularly varying at infinity we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann-Liouville process. We specialize our result for five particular counting processes. Also, we investigate H\"{o}lder continuity of the limit processes for general shot noise processes.