The $\lambda$-invariant measures of subcritical Bienaymé--Galton--Watson processes

Abstract

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, i.e.\ 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.

16 pages. Title changed in v2. Some details added

References in corpus (2)

• Quasi-stationary distributions and population processes
• On trees invariant under edge contraction