## 7Ergodic Theorems

*X*_{n}, *n* ≥ 0, is said to be a stationary sequence if for each *k* ≥ 1 it has the same distribution as the shifted sequence *X*_{n+k}, *n* ≥ 0. The basic fact about these sequences, called the ergodic theorem, is that if *E*|*f* (*X*_{0})| *<* ∞ then

If *X*_{n} is ergodic (a generalization of the notion of irreducibility for Markov chains) then the limit is *Ef* (*X*_{0}). Sections 7.1 and 7.2 develop the theory needed to prove the ergodic theorem. In Section 7.3, we apply the ergodic theorem to study the recurrence of random walks with increments that are stationary sequences finding remarkable generalizations of the i.i.d. case. In Section 7.4, we prove a subadditive ...