  For any discrete target distribution, we exploit the connection between
Markov chains and Stein's method via the generator approach and express the
solution of Stein's equation in terms of expected hitting time. This yields new
upper bounds of Stein's factors in terms of the parameters of the Markov chain,
such as mixing time and the gradient of expected hitting time. We compare the
performance of these bounds with those in the literature, and in particular we
consider Stein's method for discrete uniform, binomial, geometric and
hypergeometric distribution. As another application, the same methodology
applies to bound expected hitting time via Stein's factors. This article
highlights the interplay between Stein's method, modern Markov chain theory and
classical fluctuation theory.
