A note on mixed Poisson distributions

TL;DR

This paper explores properties of mixed Poisson distributions, including their convergence to the mixing distribution and a central limit theorem with moment convergence, enhancing understanding of their asymptotic behavior.

Contribution

It provides new insights into the convergence properties and a central limit theorem for mixed Poisson distributions, which were not previously detailed.

Findings

Mixed Poisson distributions converge to their mixing distribution as the scaling parameter tends to infinity.

A central limit theorem is established after centering by the mixing random variable.

Moment convergence results are also derived for these distributions.

Abstract

In this note we discuss additional properties of mixed Poisson distributions. We discuss the convergence of mixed Poisson distributions to its mixing distribution for the scaling parameter tending to infinity. Moreover, we obtain a central limit theorem after centering by its mixing random variable, together with moment convergence.