Steady-state Analysis and The Time-inhomogeneous Markov Chain
A Markov chain need not necessarily be time-homogeneous to have an equilibrium distribution. If there is a probability distribution over states such that
for every state j and every time n then is an equilibrium distribution of the Markov chain. Such can occur in Markov chain Monte Carlo (MCMC) methods in situations where a number of different transition matrices are used, because each is efficient for a particular kind of mixing, but each matrix respects a shared equilibrium distribution.
Read more about this topic: Markov Chain, Markov Chains, Steady-state Analysis and Limiting Distributions
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