hw8 - L 2 iF and only iF ∑ ∞ n =1 b 2 n< ∞ 4 In...

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Math 280A, Fall 2011 Homework 8 1. In this problem, the random variables X 1 , X 2 , . . . and X are defned on a probability space (Ω , F , P ) with the special property that Ω has only fnitely many points, say Ω = { ω 1 , ω 2 , . . ., ω N } , and that each singleton { ω k } is an element oF F with P [ { ω k } ] > 0 For each k . Show that X n P X iF and only iF X n X a.s. 2. Suppose that X n P X and Y n P Y as n → ∞ . (a) Let f : R R be a continuous Function. Show that f ( X n ) P f ( X ). (b) Show that X n + Y n P X + Y and that X n Y n P XY . 3. Let X 1 , X 2 , . . . be independent and identically distributed random variables, with E [ X n ] = 0 and Var[ X n ] = 1. Let { b n } be a sequence oF real numbers. Show that n =1 b n X n converges in
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Unformatted text preview: L 2 iF and only iF ∑ ∞ n =1 b 2 n < ∞ . 4. In this problem U 1 , U 2 , . . ., are independent and identically distributed random vari-ables, each uniFormly distributed on the inteval [0 , 1]. (a) Compute E [log U k ]. (b) Defne T n := ( U 1 U 2 ··· U n ) 1 /n . Show that T n P → e-1 . [Hint: Take logs, use (a) and the WLLN.] 5. Let X 1 , X 2 , . . . be independent random variables, with X n ≥ 0 For each n . Show that ∑ n X n < ∞ a.s. iF and only iF ∑ n E [min( X n , 1)] < ∞ ....
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