How singular are moment generating functions?

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.