Refined total variation bounds in the multivariate and compound Poisson approximation

TL;DR

This paper introduces improved total variation bounds for approximating convolutions of probability measures with (compound) Poisson distributions, enhancing accuracy and applicability in probabilistic modeling.

Contribution

It presents novel total variation bounds with better structure and demonstrates their usefulness through numerical examples and applications in Poisson process approximation.

Findings

New bounds outperform existing ones in accuracy

Bounds are applicable to multivariate and compound Poisson cases

Numerical examples confirm practical usefulness

Abstract

We consider the approximation of a convolution of possibly different probability measures by (compound) Poisson distributions and also by related signed measures of higher order. We present new total variation bounds having a better structure than those from the literature. A numerical example illustrates the usefulness of the bounds, and an application in the Poisson process approximation is given. The proofs use arguments from Kerstan (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964) 173-179) and Roos (J. Multivariate Anal. 69 (1999) 120-134) in combination with new smoothness inequalities, which could be of independent interest.