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Exponential convergence to a quasi-stationary distribution for birth–death processes with an entrance boundary at infinity

Part of: Probability theory on algebraic and topological structures Ergodic theory Markov processes

Published online by Cambridge University Press:  10 August 2022

Guoman He  and
Hanjun Zhang
Guoman He*
Affiliation:
Hunan University of Technology and Business
Hanjun Zhang*
Affiliation:
Xiangtan University
*
*Postal address: School of Science & Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computation, Hunan University of Technology and Business, Changsha, Hunan 410205, PR China. Email address: hgm0164@163.com
**Postal address: School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, PR China. Email address: hjz001@xtu.edu.cn
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Abstract

We study the quasi-stationary behavior of the birth–death process with an entrance boundary at infinity. We give by the h-transform an alternative and simpler proof for the exponential convergence of conditioned distributions to a unique quasi-stationary distribution in the total variation norm. In addition, we also show that starting from any initial distribution the conditional probability converges to the unique quasi-stationary distribution exponentially fast in the $\psi$ -norm.

Keywords

Birth–death processesquasi-stationary distributionh-transformrate of convergence

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60B10: Convergence of probability measures 37A25: Ergodicity, mixing, rates of mixing
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 983 - 991
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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