On distributional properties of perpetuities

TL;DR

This paper investigates the distributional characteristics of perpetuities, establishing their possible types, criteria for moments, and providing formulas for the convergence of their moment generating functions, supported by illustrative examples.

Contribution

It introduces a new approach to classify perpetuity distributions by their type and derives criteria for moments and moment generating function convergence.

Findings

Perpetuity distributions are of pure type: degenerate, absolutely continuous, or singular.

Criteria for finiteness of p-moments and exponential moments are established.

A formula for the abscissa of convergence of the moment generating function is provided.

Abstract

We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of $p$-moments, $p>0$ as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article.