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## 7Ergodic Theorems

Xn, n ≥ 0, is said to be a stationary sequence if for each k ≥ 1 it has the same distribution as the shifted sequence Xn+k, n ≥ 0. The basic fact about these sequences, called the ergodic theorem, is that if E|f (X0)| < ∞ then

If Xn is ergodic (a generalization of the notion of irreducibility for Markov chains) then the limit is Ef (X0). 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 ...

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