O T O Statistics and Probability 11. Let 1.2.... be a sequence of iid zero-mean random variables. Let (a,) be a sequence of positive constants such that E a, = 1. Define the sequence of random variables Xo, X1, X2, ... via the recursion Xo = (0, X, = (1 - a,)X-1 + a, En n= 1, 2,... (a) Show that the collection {4;} 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- a)y) < af(x) + (1 - a)f(y).
O T O Statistics and Probability 11. Let 1.2.... be a sequence of iid zero-mean random variables. Let (a,) be a sequence of positive constants such that E a, = 1. Define the sequence of random variables Xo, X1, X2, ... via the recursion Xo = (0, X, = (1 - a,)X-1 + a, En n= 1, 2,... (a) Show that the collection {4;} 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- a)y) < af(x) + (1 - a)f(y).
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Counting And Probability
Section9.4: Expected Value
Problem 20E
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