Inequalities for Branching Processes

Abstract

A branching process is considered for which the conditional distributions of the litter sizes, given the past, are allowed to vary from period to period and are required only to belong to some set $\mathscr{M}$. The process is non-Markovian in general. For various interesting $\mathscr{M}$, bounds are derived on (i) the probability of extinction, (ii) the mean time to extinction, (iii) the probability that a generation size exceeds a given number, (iv) the expected maximum generation size, and (v) the mean total population size. In (i), (ii) and (v), the optimal strategies which achieve the bounds are identified. The techniques used are similar to those used in the theory of gambling as developed by Dubins and Savage (How to Gamble if You Must, McGraw-Hill (1965)).