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

Abstract

With $X^{\ast }$ denoting a random variable with the $X$-size bias distribution, what are all distributions for $X$ such that it is possible to have $X^{\ast }=X+Y$, $Y\ge 0$, 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

Cited by in corpus (6)

• Cascades: A view from Audience
• On moderate deviations in Poisson approximation
• Distributional transformations without orthogonality relations
• Population genetic consequences of the Allee effect and the role of offspring-number variation
• Zeros of random tropical polynomials, random polytopes and stick-breaking
• An Accurate Method for Determining the Pre-Change Run-Length Distribution of the Generalized Shiryaev--Roberts Detection Procedure