  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, 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 nor
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. The approach together with the identities easily extends to discrete
time Markov chains.
