The asymptotic tails of limit distributions of continuous-time Markov chains
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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.
Original language | English |
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Journal | Advances in Applied Probability |
Volume | 56 |
Issue number | 2 |
Pages (from-to) | 693–734 |
ISSN | 0001-8678 |
DOIs | |
Publication status | Published - 2024 |
Bibliographical note
Publisher Copyright:
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust.
- Discrete-time Markov chain, quasi-stationary distribution, stationary measure, stochastic reaction network, tail distribution
Research areas
ID: 390284738