Ergodic Theory with an Application to Fractals

We have already seen examples of limit theorems in Sections V and VI that assert the convergence of temporal averages to spatial averages. Thus if x is a positive recurrent state of an aperiodic irreducible Markov chain, then $$\mathop {\lim }\limits_{n \to \infty } {N_n}(x)/n = \pi (x)$$. That is, the temporal average fraction of time the chain spends at state x converges to the spatial average π(x). Similarly, for Markov jump processes $$\frac{1}{t}\int_0^t {{I_{\{ x\} }}(X(s))ds \to \pi (x)} $$ when x is positive recurrent.