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

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.

30 pages