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Unformatted text preview: P ( a Y n b ) P ( a Z b ). (We wont prove this.) We are going to let Y n = ( S nn ) / n . Let W i = ( X i ) / . Then E W i = 0, Var W i = Var X i 2 = 1, the W i are independent, and S nn n = n i =1 W i n . So there is no loss of generality in assuming that = 0 and = 1. Then m Y n ( t ) = E e tY n = E e ( t/ n )( S n ) = m S n ( t/ n ) . Since the X i are i.i.d., all the X i have the same moment generating function. Since S n = X 1 + + X n , then m S n ( t ) = m X 1 ( t ) m X n ( t ) = [ m X 1 ( t )] n . If we expand e tX 1 as a power series, m X 1 ( t ) = E e tX 1 = 1 + t E X 1 + t 2 2! E ( X 1 ) 2 + t 3 3! E ( X 1 ) 3 + . 47...
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This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.
 Spring '10
 ansan

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