This paper investigates tail asymptotics of stationary distributions and
quasi-stationary 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 continuous-time 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