No4(2004) - MATH 587 Assignment 4 Due November 30, 2004 (1)...

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MATH 587 Assignment 4 Due November 30, 2004 (1) Let X L 2 (Ω , F ,P ) and let G be a σ -subalgebra of F . Show that among all G -measurable r.v.’s Y L 2 (Ω , F ), there is a unique (a.s.) r.v. Y 0 which minimizes k X Y k 2 . (2) (a) Suppose that Y 1 ,Y 2 ,... are non-negative random variables and that n =1 EY n < . Show that as n →∞ , Y n 0 a.s. (b) Use (a) to show that if X 1 ,X 2 is a sequence of iid random variables with mean µ , variance σ 2 , and a Fnite fourth moment, then S n /n µ a.s. as n . (Hint: put Y n =( S n ) 4 /n 4 , where S n = n i =1 X i .) This is due to Cantelli.
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