Open Access
June, 1973 Inequalities for Branching Processes
Bruce W. Turnbull
Ann. Probab. 1(3): 457-474 (June, 1973). DOI: 10.1214/aop/1176996939


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)).


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Bruce W. Turnbull. "Inequalities for Branching Processes." Ann. Probab. 1 (3) 457 - 474, June, 1973.


Published: June, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0258.60063
MathSciNet: MR353477
Digital Object Identifier: 10.1214/aop/1176996939

Primary: 60J80
Secondary: 60G40 , 60G45

Keywords: branching processes , Chebyshev-like inequalities , dynamic programming , gambling theory , Martingales , non-Markovian processes , stopping times

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 3 • June, 1973
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