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# sec05 - Large Sample Theory Ferguson Exercises Section 5...

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Large Sample Theory Ferguson Exercises, Section 5, Central Limit Theorems. 1. (a) Using a Chebyshev’s-Inequality-like argument, show that (assuming the expec- tations exist) E | X | 2+ α t α E[ X 2 I( | X | ≥ t )] for all α > 0 and t > 0. (b) Using part (a) and Lindeberg, prove Liapounov’s Theorem: Let X n 1 , X n 2 , . . . , X nn be independent with E X nj = 0 and E | X nj | 2+ α < for some α > 0 and all n and j . Let Z n = n j =1 X nj and B 2 n = Var Z n = n j =1 Var X nj . Then Z n /B n L −→ N (0 , 1), provided 1 B 2+ α n n j =1 E | X nj | 2+ α 0 as n → ∞ . 2. Let X 1 , X 2 , . . . be independent exponential random variables with means β 1 , β 2 , . . . respectively, and let Z n = X 1 + · · · + X n . Show that if max 1 j n β 2 j / n j =1 β 2 j 0 as n → ∞ , then ( Z n E Z n ) / Var Z n L −→ N (0 , 1). (Use Liapounov’s Theorem with α = 2.) 3. (a) Let X 1 , X 2 , . . . be independent Poisson random variables with means λ 1 , λ 2 , . . . respectively, and let Z n = X 1 + · · · + X n . Show that ( Z n E Z n ) / Var Z n L −→ N (0 , 1) if and only if n 1 λ j → ∞ . (b) Show that this can provide an example to show you can get asymptotic normality without the Lindeberg condition being satisfied. 4. As an illustration of the use of Kendall’s tau, here is a famous little example taken from M. G. Kendall’s 1948 book, Rank Correlation Methods . Suppose a number of boys are ranked according to their ability in mathematics and music. Such a pair of rankings

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