Conditions under which a Markov Chain Converges to its Steady State in Finite Time

P. W. Glynn and D. L. Iglehart

Probability in the Engineering and Informational Sciences, Vol. 2, 377-382 (1988)

Analysis of the initial transient problem of Monte Carlo steady-state simulation motivates the following question for Markov chains: when does there exist a deterministic T such that P{X(T) = y|X(0) = x} = π(y), where π is the stationary distribution of X? We show that this can essentially never happen for a continuous-time Markov chain; in discrete time, such processes are i.i.d. provided the transition matrix is diagonalizable.