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The asymptotic tails of limit distributions of continuous-time Markov chains

Part of: Markov processes

Published online by Cambridge University Press:  06 October 2023

Chuang Xu Open the ORCID record for Chuang Xu [Opens in a new window] ,
Mads Christian Hansen  and
Carsten Wiuf
Chuang Xu*
Affiliation:
University of Hawai’i at Mānoa
Mads Christian Hansen*
Affiliation:
University of Copenhagen
Carsten Wiuf*
Affiliation:
University of Copenhagen
*
*Postal address: Department of Mathematics, University of Hawai’i at Mānoa, Honolulu, HI 96822, USA. Email address: chuangxu@hawaii.edu
**Postal address: Department of Mathematical Sciences, University of Copenhagen, Copenhagen, 2100, Denmark.
**Postal address: Department of Mathematical Sciences, University of Copenhagen, Copenhagen, 2100, Denmark.
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Abstract

This paper investigates tail asymptotics of stationary distributions and quasi-stationary distributions (QSDs) of continuous-time Markov chains on subsets of the non-negative integers. Based on the so-called flux-balance equation, we establish identities for stationary measures and QSDs, which we use to derive tail asymptotics. In particular, for continuous-time Markov chains with asymptotic power law transition rates, tail asymptotics for stationary distributions and QSDs are classified into three types using three easily computable parameters: (i) super-exponential distributions, (ii) exponential-tailed distributions, and (iii) sub-exponential distributions. Our approach to establish tail asymptotics of stationary distributions is different from the classical semimartingale approach, and we do not impose ergodicity or moment bound conditions. In particular, the results also hold for explosive Markov chains, for which multiple stationary distributions may exist. Furthermore, our results on tail asymptotics of QSDs seem new. We apply our results to biochemical reaction networks, a general single-cell stochastic gene expression model, an extended class of branching processes, and stochastic population processes with bursty reproduction, none of which are birth–death processes. Our approach, together with the identities, easily extends to discrete-time Markov chains.

Keywords

Discrete-time Markov chainstationary measuretail distributionquasi-stationary distributionstochastic reaction network

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J28: Applications of continuous-time Markov processes on discrete state spaces

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 56 , Issue 2 , June 2024 , pp. 693 - 734
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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