Factorization Property of Generalized s-self-decomposable measures and class $L^f$ distributions

TL;DR

This paper explores the factorization property of generalized s-self-decomposable and selfdecomposable distributions using the random integral representation method, providing new insights into their structure.

Contribution

It introduces a random integral representation for these classes, enhancing understanding of their factorization properties and extending previous results.

Findings

Representation for classes _{e} and L^f derived

New characterization of factorization property established

Connections to generalized s-selfdecomposable measures clarified

Abstract

The method of \emph{random integral representation}, that is, the method of representing a given probability measure as the probability distribution of some random integral, was quite successful in the past few decades. In this note we will find such a representation for generalized s-selfdecomposable and selfdecomposable distributions that have the \emph{factorization property}. These are the classes $\mathcal{U}^f_{\be}$ and $L^f$, respectively.