Journal of Applied Probability

On largest offspring in a critical branching process with finite variance

Jean Bertoin

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Abstract

Continuing the work in Bertoin (2011) we study the distribution of the maximal number X*k of offspring amongst all individuals in a critical Galton‒Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail F̅ with index −α for α>2 (and, hence, finite variance). We show that X*k suitably normalized converges in distribution to a Fréchet law with shape parameter α/2; this contrasts sharply with the case 1< α<2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

Article information

Source
J. Appl. Probab., Volume 50, Number 3 (2013), 791-800.

Dates
First available in Project Euclid: 5 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.jap/1378401236

Digital Object Identifier
doi:10.1239/jap/1378401236

Mathematical Reviews number (MathSciNet)
MR3188595

Zentralblatt MATH identifier
1277.60045

Subjects
Primary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Branching process maximal offspring extreme value theory Cox process

Citation

Bertoin, Jean. On largest offspring in a critical branching process with finite variance. J. Appl. Probab. 50 (2013), no. 3, 791--800. doi:10.1239/jap/1378401236. https://projecteuclid.org/euclid.jap/1378401236


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