  In this paper, we are interested in investigating the perturbation bounds for
the stationary distributions for discrete-time or continuous-time Markov chains
on a countable state space. For discrete-time Markov chains, two new norm-wise
bounds are obtained. The first bound is rather easy to be obtained since the
needed condition, equivalent to uniform ergodicity, is imposed on the
transition matrix directly. The second bound, which holds for a general
(possibly periodic) Markov chain, involves finding a drift function. This drift
function is closely related with the mean first hitting times. Some
$V$-norm-wise bounds are also derived based on the results in Kartashov (1986).
Moreover, we show how the bounds developed in this paper and one bound given in
Seneta (1988) can be extended to continuous-time Markov chains. Several
examples are shown to illustrate our results or to compare our bounds with the
known ones in the literature.
