Crossings of smooth shot noise processes

TL;DR

This paper analyzes the level crossings of smooth shot noise processes, deriving an integral formula for their mean crossings and demonstrating Gaussian convergence in high-intensity regimes, with detailed study of Gaussian kernels.

Contribution

It provides a new integral formula for the expected number of crossings of smooth shot noise processes and establishes Gaussian convergence as the process intensity increases.

Findings

Derived an integral formula for mean number of crossings.

Proved Gaussian convergence in high-intensity regimes.

Analyzed Gaussian kernel functions in detail.

Abstract

In this paper, we consider smooth shot noise processes and their expected number of level crossings. When the kernel response function is sufficiently smooth, the mean number of crossings function is obtained through an integral formula. Moreover, as the intensity increases, or equivalently, as the number of shots becomes larger, a normal convergence to the classical Rice's formula for Gaussian processes is obtained. The Gaussian kernel function, that corresponds to many applications in physics, is studied in detail and two different regimes are exhibited.