A stationary distribution of a Markov chain with transition matrix 
is a probability vector 
satisfying
 |
If the initial state has distribution 
, then every later state has the same distribution. Thus a
stationary distribution is a left eigenvector of
with eigenvalue 1, normalized to have nonnegative entries summing to 1.
A finite Markov chain in which every state is reachable from every other has a unique stationary distribution with all entries positive. Convergence to it from every initial distribution additionally requires that the greatest common divisor of the possible state-return times be 1.
