Regularity of the laws of shot noise series and of related processes

TL;DR

This paper studies the regularity properties of shot noise series and Poisson integrals, providing conditions for their absolute continuity and total variation continuity, especially focusing on truncated series.

Contribution

It introduces a new method based on disintegration of the probability space conditioned on the first jumps of the Poisson process to analyze regularity.

Findings

Conditions for absolute continuity of shot noise series laws.

Conditions for total variation continuity of Poisson integrals.

Analysis of truncated series cases.

Abstract

We investigate the regularity of shot noise series and of Poisson integrals. We give conditions for the absolute continuity of their law with respect to Lebesgue measure and for their continuity in total variation norm. In particular, the case of truncated series in adressed. Our method relies on a disintegration of the probability space based on a mere conditioning by the first jumps of the underlying Poisson process.