The $\lambda$-invariant measures of subcritical Bienaym\'e--Galton--Watson processes

TL;DR

This paper provides an explicit integral formula for $mbda$-invariant measures of subcritical Bienayme9--Galton--Watson processes killed at extinction, extending previous results and offering elementary proofs without Martin boundary theory.

Contribution

It introduces a new integral representation for $mbda$-invariant measures, characterizing all quasi-stationary distributions of these processes with elementary methods.

Findings

Explicit integral formula for $mbda$-invariant measures

Characterization of all quasi-stationary distributions

Elementary proofs avoiding Martin boundary theory

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 Bienaym\'e--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.