A New Factorization Property of the Selfdecomposable Probability Measures

TL;DR

This paper introduces a new factorization property of selfdecomposable probability measures, characterizes the class of distributions with this property, and explores its algebraic structure and examples, including the Lévy stochastic area integral.

Contribution

It establishes the factorization property for selfdecomposable measures, characterizes the class of such distributions, and studies its algebraic and structural properties.

Findings

The convolution of a selfdecomposable distribution with its background law is selfdecomposable iff the background law is s-selfdecomposable.

The class of distributions with the factorization property, L^f, has a rich algebraic structure.

Examples include the Lévy stochastic area integral and a nested family of subclasses within L^f.

Abstract

We prove that the convolution of a selfdecomposable distribution with its background driving law is again selfdecomposable if and only if the background driving law is s-selfdecomposable. We will refer to this as the \textit{factorization property} of a selfdecomposable distribution; let $L^f$ denote the set of all these distributions. The algebraic structure and various characterizations of $L^f$ are studied. Some examples are discussed, the most interesting one being given by the L\'evy stochastic area integral. A nested family of subclasses $L^{f}_n, n\ge 0,$ (or a filtration) of the class $L^f$ is given.