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m157cltproof

# m157cltproof - Proof of Central Limit Theorem H Krieger...

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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 0 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 π Z x -∞ e - u 2 / 2 du. Proof: Let ˆ F be the characteristic function of the common distribution of the { X n } . Then for every t R , the characteristic function of S n /

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m157cltproof - Proof of Central Limit Theorem H Krieger...

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