Markov Chains: Analytic and Monte Carlo Computations

3.2 Invariant measures and recurrence

3.2.1 Invariant laws and measures

3.2.1.1 Invariant laws, stationary chain, and balance equations

Let be a Markov chain with matrix , and be its instantaneous laws. Then, solves the linear (or affine) recursion and, under weak continuity assumptions, can converge to some law only if is a fixed point for the recursion, and hence only if .

If a law is s.t. , and if , then

and hence, is a Markov chain with matrix and initial law and thus has same law ...