1. Let X1, X2,... , be i.i.d. real random variables with E |X1| < >∞ and Sn := X1 +· · · + Xn . Verify that |Sn | is a subadditive sequence and find the limit of |Sn |/n in terms of an integral for X1.

2. Let (X, A, P) be a probability space and T a measure-preserving trans- formation of X onto itself. Let Y be a real random variable on X and Yj := Y ? T j for j = 1, 2,... . Let S0 := 0 and for n = 1, 2,... , let Sn := Y1 + ··· + Yn and fn := max{Sk - Sj :0 ≤ j ≤ k ≤ n}. Prove that { fn } is subadditive.