Selfdecomposability, perpetuity and stopping times

TL;DR

This paper explores the connection between selfdecomposable distributions, stopping times, and perpetuity laws, demonstrating that all selfdecomposable distributions can be represented as perpetuities, with implications for stochastic analysis in Banach spaces.

Contribution

It introduces a novel approach to incorporate stopping times into the theory of selfdecomposable distributions using random integrals, and shows all such distributions are perpetuities.

Findings

Selfdecomposable distributions are perpetuities.

Stopping times can be used in the analysis of limit distributions.

Applications to Banach space-valued random variables.

Abstract

In the probability theory limit distributions (or probability measures) are often characterized by some convolution equations (factorization properties) rather than by Fourier transforms (the characteristic functionals). In fact, usually the later follows the first one. Equations, in question, involve the multiplication by the positive scalars $c$ or an action of the corresponding dilation $T_{c}$ on measures. In such a setting, it seems that there is no way for stopping times (or in general, for the stochastic analysis) to come into the "picture". However, if one accepts the view that the primary objective, in the classical limit distributions theory, is to describe the limiting distributions (or random variables) by the tools of random integrals/functionals then one can use the stopping times. In this paper we illustrate such a possibility in the case of selfdecomposability random variables (i.e. L\'evy class L) with values in a real separable Banach space. Also some applications of our approach to perpetuity laws are presented; cf. \cite{Du1}, \cite{ED1}, \cite{GG1}. In fact, we show that all selfdecomposable distributions are perpetuity laws.