  We study perturbation theory and uniform ergodicity for discrete-time Markov
chains on general state spaces in terms of the uniform moments of the first
hitting times on some set. The methods we adopt are different from previous
ones. For reversible and non-negative definite Markov chains, we first
investigate the geometrically ergodic convergence rates. Based on the
estimates, together with a first passage formula, we then get the convergence
rates in uniform ergodicity. If the transition kernel $P$ is only reversible,
we transfer to study the two-skeleton chain with the transition kernel $P^2$.
At a technical level, the crucial point is to connect the geometric moments of
the first return times between $P$ and $P^2$.
