hw7 - smallest such constant That is b X b ∞:= inf x ≥...

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Math 280A, Fall 2011 Homework 7 1. Problem 1 (page 260) 2. Let x 1 , x 2 , . . ., x n be strictly positive numbers. Show that 1 n n s k =1 x k p n P k =1 x k ± 1 /n , with equality only if x 1 = x 2 = ··· = x n . [Hint: The natural logarithm log is a strictly concave function.] 3. Let X be a random variable and let a < b be real numbers such that the moment generating function ϕ ( t ) := E [exp( tX )] is Fnite for all t in the interval ( a, b ). (a) Use H¨ older’s inequality to show that ϕ ( λs + (1 λ ) t ) ϕ ( s ) λ · ϕ ( t ) 1 - λ for all 0 < λ < 1 and all s, t ( a, b ). (b) Deduce from (a) that f ( t ) := log ϕ ( t ) is a convex function on the interval ( a, b ). 4. Recall from class that a random variable X is essentially bounded ( X L ) if there is a constant M such that P [ | X | ≤ M ] = 1. The L norm of such a random variable is the
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Unformatted text preview: smallest such constant. That is, b X b ∞ := inf { x ≥ 0 : P [ | X | ≤ x ] = 1 } = sup { x ∈ R : P [ | X | > x ] = 0 } . (a) Show that if X ∈ L 1 and Y ∈ L ∞ then b XY b 1 ≤ b X b 1 · b Y b ∞ . (b) Show that if X ∈ L ∞ and Y ∈ L ∞ then b X + Y b ∞ ≤ b X b ∞ + b Y b ∞ . 5. Let X 1 , X 2 , . . . be identically distributed random variables (not necessarily independent) such that E [ X 2 1 ] < ∞ . (a) Show that lim n →∞ n P [ | X 1 | > ǫ √ n ] = 0 for each ǫ > 0. (b) Show that max( X 1 , X 2 , . . ., X n ) / √ n P → 0 as n → ∞ ....
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This note was uploaded on 12/11/2011 for the course MATH 280a taught by Professor Driver,b during the Fall '08 term at UCSD.

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