This paper develops bounds on the distribution function of the empirical mean for irreducible finite-state Markov chains. One approach, explored by D. 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 D. Gillman and extend it to non-reversible 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.