An unexpected connection between branching processes and optimal stopping

Abstract

A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process $Z_n$ with offspring distribution $Y$, there exists a random variable $X$ such that the probability $P(Z_n=0)$ of extinction of the $n$th generation in the branching process equals the value obtained by optimally stopping the sequence $X_1,...,X_n$, where these variables are i.i.d distributed as $X$. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple `stopping rule' method for computing various characteristics of branching processes, including rates of convergence of the $n^{th}$ generation's extinction probability to the eventual extinction probability, for the supercritical, critical and subcritical Galton-Watson process. Examples, bounds, further generalizations and a connection to classical prophet inequalities are presented. Throughout, the aim is to show how this unexpected connection can be used to translate methods from one area of applied probability to another, rather than to provide the most general results.