09831HW4solutions

# 09831HW4solutions - HOMEWORK 4 SOLUTIONS Problem 1 First...

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HOMEWORK 4 SOLUTIONS Problem 1 First observe that (1) n - 1 E S n = n - 1 X 1 k n E X n,k = X 1 k n n - 1 u ± k n ² which in particular is the Riemann sum of u for the partition ³ k n : 1 k n ´ . Since u is integrable, we know that that the Riemann sum converges to the integral of u , therefore (2) n - 1 E S n n →∞ Z 1 0 u ( x ) dx. Deﬁne ξ n,i = X n,i - E X n,i . Now consider the fourth moment of S n - E S n n = n - 1 n X i =1 ξ n,i . We estimate the fourth moment: n - 4 E X 1 i n ξ n,i 4 = n - 4 E X α 1 + ... + αn =4 a i 0 Y 1 i n ξ α i n,i = n - 4 X α 1 + ... + αn =4 a i 0 Y 1 i n E ξ α i n,i (3) where the last equality follows from the independence (in i ) of the ξ n,i . If at least one of the α i is 1, the product becomes 0, since the ξ n,i ’s are centered. Then the possible terms that show up in the sum (3) are: ξ 2 n,i ξ 2 n,j . There are exactly

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## This note was uploaded on 11/18/2009 for the course MATH 241 taught by Professor Reed during the Spring '09 term at Duke.

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09831HW4solutions - HOMEWORK 4 SOLUTIONS Problem 1 First...

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