Size bias, sampling, the waiting time paradox, and infinite divisibility: when is the increment independent?

TL;DR

This paper characterizes distributions where size biasing leads to an independent increment, linking size bias, infinite divisibility, and various probabilistic phenomena like the waiting time paradox and renewal theory.

Contribution

It provides a complete characterization of distributions allowing an independent increment after size biasing, extending Steutel's results and connecting to multiple areas in probability theory.

Findings

Identifies all distributions where size biasing yields an independent increment.

Links size biasing to infinite divisibility and the waiting time paradox.

Discusses implications for renewal theory, sampling, and lognormal distributions.

Abstract

With $X^*$ denoting a random variable with the $X$-size bias distribution, what are all distributions for $X$ such that it is possible to have $X^*=X+Y$, $Y\geq 0$, with $X$ and $Y$ {\em independent}? We give the answer, due to Steutel \cite{steutel}, and also discuss the relations of size biasing to the waiting time paradox, renewal theory, sampling, tightness and uniform integrability, compound Poisson distributions, infinite divisibility, and the lognormal distributions.