11. Let $1, 52,. .. be a sequence of iid zero-mean random variables. Let (a,) be a sequence of positive constants such that _ _ @, = 1. Define the sequence of random variables Xo, X1, X2, . . . via the recursion Xo = 0, Xn = (1 - an)Xn-1+ an5n, n = 1,2, ... (a) Show that the collection { } is uniformly integrable. (b) Use proposition 3.36 to show that {X,} is uniformly integrable. Hint: For a convex function f : R - R and a e (0, 1), f(ax + (1 - @)y) saf(x) + (1- () f ( y).Proposition 3.36: Conditions for Uniform Integrability Let K 0, 5. K is UI if and only if (a) supxex E[X| 0 there is a 6 > 0 such that for every event H: (3.37) P(H)