If Xn, n 1, are i.i.d. r.v.s defined on the probability space (, A, P), then (i) Show that the process X = (X1,

If Xn, n ≥ 1, are i.i.d. r.v.s defined on the probability space (Ω, A, P), then
(i) Show that the process X = (X1, X2, ...) is ergodic.
(ii) Derive the Strong Law of Large Numbers under the assumption that ɛX1 is finite.
(i) By Proposition 3, it follows that X = (X1, X2,...) is stationary. Let J be the σ-field of invariant sets relative to X (see Definition 11), and let T be the tail σ-field defined on X = (X1. X2,...) (see Definition 1 and preceding discussion in Chapter 14). Then J ( T. Conclude the discussion by using Theorem 10 in Chapter 14.
(ii) Refer to Corollary 2 of Theorem 3.