How singular are moment generating functions?

TL;DR

This paper investigates the boundary singularities of moment generating functions of finite measures, analyzing specific examples and discussing regularity issues through tensorization of distributions.

Contribution

It provides a detailed analysis of singularities in moment generating functions and introduces a method to address regularity problems via tensorization of distributions.

Findings

Identifies types of singularities at the boundary of MGF domains

Elaborates on Example 7.3 from Barndorff-Nielsen's book

Proposes a solution to regularity issues through tensorization

Abstract

This short note concerns the possible singular behaviour of moment generating functions of finite measures at the boundary of their domain of existence. We look closer at Example 7.3 in O. Barndorff-Nielsen's book "Information and Exponential Families in Statistical Theory (1978)" and elaborate on the type of exhibited singularity. Finally, another regularity problem is discussed and it is solved through tensorizing two Barndorff- Nielsen's distributions.