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February 2018 The $\lambda$-invariant measures of subcritical Bienaymé–Galton–Watson processes
Pascal Maillard
Bernoulli 24(1): 297-315 (February 2018). DOI: 10.3150/16-BEJ877


A $\lambda$-invariant measure of a sub-Markov chain is a left eigenvector of its transition matrix of eigenvalue $\lambda$. In this article, we give an explicit integral representation of the $\lambda$-invariant measures of subcritical Bienaymé–Galton–Watson processes killed upon extinction, that is, upon hitting the origin. In particular, this characterizes all quasi-stationary distributions of these processes. Our formula extends the Kesten–Spitzer formula for the (1-)invariant measures of such a process and can be interpreted as the identification of its minimal $\lambda$-Martin entrance boundary for all $\lambda$. In the particular case of quasi-stationary distributions, we also present an equivalent characterization in terms of semi-stable subordinators.

Unlike Kesten and Spitzer’s arguments, our proofs are elementary and do not rely on Martin boundary theory.


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Pascal Maillard. "The $\lambda$-invariant measures of subcritical Bienaymé–Galton–Watson processes." Bernoulli 24 (1) 297 - 315, February 2018.


Received: 1 December 2015; Revised: 1 June 2016; Published: February 2018
First available in Project Euclid: 27 July 2017

zbMATH: 06778329
MathSciNet: MR3706758
Digital Object Identifier: 10.3150/16-BEJ877

Keywords: Bienaymé–Galton–Watson process , invariant measure , Martin boundary , quasi-stationary distribution , Schröder equation , semi-stable process

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 1 • February 2018
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