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The \(\lambda\)-invariant measures of subcritical Bienaymé-Galton-Watson processes. (English) Zbl 1454.60133
Summary: 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.

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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