Factorization property of generalized s-selfdecomposable measures and class $L^f$ distributions$^1$

TL;DR

This paper explores the random integral representations of generalized s-selfdecomposable and selfdecomposable distributions with the factorization property, advancing understanding of their structure and classification.

Contribution

It provides new random integral representations for classes _{e} and L^f with the factorization property, enriching the theory of these distributions.

Findings

Derived integral representations for _{e} and L^f distributions.

Characterized the factorization property within these classes.

Enhanced the theoretical framework of selfdecomposable measures.

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