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Quasi-stationary distributions for subcritical branching Markov chains

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  10 September 2025

Wenming Hong  and
Dan Yao Open the ORCID record for Dan Yao [Opens in a new window]
Wenming Hong*
Affiliation:
Beijing Normal University
Dan Yao*
Affiliation:
Beijing Normal University
*
*Postal address: School of Mathematical Sciences and Laboratory of Mathematics and Complex Systems, Beijing 100875, P.R. China.
*Postal address: School of Mathematical Sciences and Laboratory of Mathematics and Complex Systems, Beijing 100875, P.R. China.
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Abstract

Consider a subcritical branching Markov chain. Let $Z_n$ denote the counting measure of particles of generation n. Under some conditions, we give a probabilistic proof for the existence of the Yaglom limit of $(Z_n)_{n\in\mathbb{N}}$ by the moment method, based on the spinal decomposition and the many-to-few formula. As a result, we give explicit integral representations of all quasi-stationary distributions of $(Z_n)_{n\in\mathbb{N}}$, whose proofs are direct and probabilistic, and do not rely on Martin boundary theory.

Keywords

Branching Markov chainYaglom limitquasi-stationary distributionconditional reduced Galton–Watson process

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems
Secondary: 60F10: Large deviations

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 58 , Issue 1 , March 2026 , pp. 425 - 471
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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