A calculus on L\'evy exponents and selfdecomposability on Banach spaces

TL;DR

This paper introduces a new calculus for Lévy exponents in infinite-dimensional Banach spaces, enabling a better understanding of infinitely divisible measures and their selfdecomposability properties.

Contribution

It develops a calculus on Lévy exponents based on random integrals, and shows that every selfdecomposable measure can be factorized into a selfdecomposable measure and an s-selfdecomposable background measure.

Findings

Established a calculus for Lévy exponents in Banach spaces

Proved factorization of selfdecomposable measures into specific components

Extended previous results on selfdecomposability in probability measures

Abstract

In infinite dimensional Banach spaces there is no complete characterization of the L\'evy exponents of infinitely divisible probability measures. Here we propose \emph{a calculus on L\'evy exponents} that is derived from some random integrals. As a consequence we prove that \emph{each} selfdecomposable measure can by factorized as another selfdecomposable measure and its background driving measure that is s-selfdecomposable. This complements a result from the paper of Iksanov-Jurek-Schreiber in the Annals of Probability \textbf{32}, 2004.}