The Annals of Probability

Conceptual Proofs of $L$ Log $L$ Criteria for Mean Behavior of Branching Processes

Russell Lyons, Robin Pemantle, and Yuval Peres

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Abstract

The Kesten-Stigum theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an $L \log L$ condition is decisive. In critical and subcritical cases, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least $n$ generations. We give conceptual proofs of these theorems based on comparisons of Galton-Watson measure to another measure on the space of trees. This approach also explains Yaglom's exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution.

Article information

Source
Ann. Probab., Volume 23, Number 3 (1995), 1125-1138.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988176

Digital Object Identifier
doi:10.1214/aop/1176988176

Mathematical Reviews number (MathSciNet)
MR1349164

Zentralblatt MATH identifier
0840.60077

JSTOR
links.jstor.org

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Galton-Watson size-biased distributions

Citation

Lyons, Russell; Pemantle, Robin; Peres, Yuval. Conceptual Proofs of $L$ Log $L$ Criteria for Mean Behavior of Branching Processes. Ann. Probab. 23 (1995), no. 3, 1125--1138. doi:10.1214/aop/1176988176. https://projecteuclid.org/euclid.aop/1176988176


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