# hw9 - ∑ ∞ k =1 X k k converges a.s and then appeal to...

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Math 280A, Fall 2011 Homework 9 1. Problem 2 (page 323) 2. Problem 4 (page 324) 3. Problem 9 (page 325) 4. Problem 17 (page 327) 5. Let A 1 ,A 2 ,... be independent events. (a) Show that, as n → ∞ , 1 n n s k =1 1 A k 1 n n s k =1 P [ A k ] P 0 . [Hint: Compute the mean and variance of the diFerence of sums above, and then use Chebyshev’s inequality.] (b) Show that, as n → ∞ , 1 n n s k =1 1 A k 1 n n s k =1 P [ A k ] 0 , a.s. [Hint: Let X k = 1 A k P [ A k ]. Show that
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Unformatted text preview: ∑ ∞ k =1 X k k converges a.s., and then appeal to Kronecker’s lemma.] 6. Let X 1 ,X 2 ,... be independent and identically distributed, with E | X 1 | < ∞ and μ := E [ X 1 ] n = 0. De±ne S n := X 1 + ··· + X n and M n := max( | X 1 | , | X 2 | ,..., | X n | ). Show that M n S n → , a.s. as n → ∞ ....
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