This paper develops bounds on the distribution function of the
empirical mean for irreducible finite-state Markov chains. One approach,
explored by Gillman, reduces this problem to bounding the largest eigenvalue of
a perturbation of the transition matrix for the Markov chain. By using
estimates on eigenvalues given in Kato's book Perturbation Theory for
Linear Operators, we simplify the proof of Gillman and extend it to
nonreversible finite-state Markov chains and continuous time. We also set out
another method, directly applicable to some general ergodic Markov kernels
having a spectral gap.