The $λ$-invariant measures of subcritical Bienaymé--Galton--Watson processes

Abstract

A $λ$-invariant measure of a sub-Markov chain is a left eigenvector of its transition matrix of eigenvalue $λ$. In this article, we give an explicit integral representation of the $λ$-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 $λ$-Martin entrance boundary for all $λ$. 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