**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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—Willa Cather (1873–1947)