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

Published online by Cambridge University Press:  10 August 2022

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: [email protected]
**Postal address: School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, PR China. Email address: [email protected]

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.

Type
Original Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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