On Boolean selfdecomposable distributions

TL;DR

This paper explores Boolean selfdecomposable distributions, establishing their regularity properties, analyzing the effects of shifting measures, and providing examples, including the normal distribution, to illustrate these concepts.

Contribution

It introduces Boolean selfdecomposable distributions, proves their regularity properties, and examines how shifting affects their selfdecomposability, with specific examples.

Findings

Number of atoms in Boolean selfdecomposable distributions is at most two

Singular continuous part of such distributions is zero

Standard normal distribution is Boolean selfdecomposable, shifted normal is not for large shifts

Abstract

This paper introduces the class of selfdecomposable distributions concerning Boolean convolution. A general regularity property of Boolean selfdecomposable distributions is established; in particular the number of atoms is at most two and the singular continuous part is zero. We then analyze how shifting probability measures changes Boolean selfdecomposability. Several examples are presented to supplement the above results. Finally, we prove that the standard normal distribution $N(0,1)$ is Boolean selfdecomposable but the shifted one $N(m,1)$ is not for sufficiently large $|m|$.