The asymptotic tails of limit distributions of continuous time Markov chains

Abstract

This paper investigates tail asymptotics of stationary distributions and quasistationary distributions of continuous-time Markov chains on a subset of the non-negative integers. A new identity for stationary measures is established. In particular, for continuoustime Markov chains with asymptotic power-law transition rates, tail asymptotics for stationary distributions are classified into three types by three easily computable parameters: (i) Conley-Maxwell-Poisson distributions (light-tailed), (ii) exponential-tailed distributions, and (iii) heavy-tailed distributions. Similar results are derived for quasi-stationary distributions. The approach to establish tail asymptotics is different from the classical semimartingale approach. We apply our results to biochemical reaction networks (modeled as continuous-time Markov chains), 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.

An Algebraic Approach to Product-form Stationary Distributions for Some Reaction Networks

“…While different results exist like e.g. for positive recurrence [3], non-explositivity of complex balanced CRN [12], extinction/absorption events [25,28], quasi-stationary distributions [28] or 1-d stochastic CRNs [37], we are far from a complete characterization for most.…”

“…A → B for an example of a trivial one). Note also that for CRNs with infinite irreducible components, positive recurrence can be hard to check, cf., e.g., [37].…”