**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

### Famous quotes containing the words chain and/or analysis:

“A *chain* is no stronger than its weakest link, and life is after all a *chain*.”

—William James (1842–1910)

“The spider-mind acquires a faculty of memory, and, with it, a singular skill of *analysis* and synthesis, taking apart and putting together in different relations the meshes of its trap. Man had in the beginning no power of *analysis* or synthesis approaching that of the spider, or even of the honey-bee; but he had acute sensibility to the higher forces.”

—Henry Brooks Adams (1838–1918)