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83Lemma 2: ErgodicityLet X = X i , i ∈ Z d be a stationary process in L p , 1 ≤ p < ∞. If (D M ) is a sequenceof bounded convex sets such that d(D M ) → ∞, and if ¯X M = |D M | −1 ∑ D MX i , thenit follows that lim M→∞ ¯X M = E(X 0 |I) in L p .I now describe a sequence of bounded convex sets (D M ) which satisfy therequirements of the lemma. Consider a sequence of rectangular subsets of aninfinitely large image, with lower left hand corner at the origin and each side oflength M. For the sequence of rectangular sets I have defined (and in fact for anysequence which increases in size in both dimensions as M increases), it is clearthat d(D M ) → ∞ as M → ∞. Furthermore, |D M | = M 2 , so ¯X M defined in thelemma is the usual sample average.Proof of Theorem 5.1I will examine each case in turn.Case 1: K T = 1The inequality in equation 5.17 can be rewritten as follows.2 log(L ˆX(Y |K)) − D K log(N) < 2 log(L ˆX(Y |K T )) − D KT log(N) (5.31)⇒ log(L ˆX(Y |K)) − (D K − D KT ) log(N)/2 < log(L ˆX(Y |K T )) (5.32)⇒ (L ˆX (Y |K)) exp(−(D K − D KT ) log(N)/2) < (L ˆX (Y |K T )) (5.33)