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Unformatted text preview: Proof of Central Limit Theorem H. Krieger, Mathematics 157, Harvey Mudd College Spring, 2005 Preliminary Inequalities: In order to utilize the result (sometimes called the continuity theorem) that convergence in distribution is equivalent to point wise convergence of the corresponding characteristic functions, we need the fol lowing estimates about Taylor expansions of exponential functions. 1. If u 0, then e u 1 + u u 2 / 2 . 2. If t is real, then  e it 1 it   t  2 / 2 and  e it 1 it ( it ) 2 / 2   t  3 / 6 . Central Limit Theorem: Let { X n } be a sequence of i.i.d. (independent identically distributed) random variables with common mean 0 and common variance 1. Then, if Z N (0 , 1) and S n = X 1 + X 2 + + X n , we have S n / n Z in distribution as n . In other words, for every x R , lim n P X 1 + X 2 + + X n n x = 1 2 Z x e u 2 / 2 du....
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This note was uploaded on 07/18/2011 for the course AEB 6933 taught by Professor Carriker during the Fall '09 term at University of Florida.
 Fall '09
 CARRIKER

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