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# Sequences and Limit Theorems

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Sequences  ::  Convergence  ::  Limit  ::  Central Limit

## Central Limit Theorems

Theorem 3   Central Limit Theorem Suppose is a sequence of i.i.d. random variables with mean and variance . Then

where

That is,

The main point: Sums of i.i.d. random variables tend to look Gaussian .

To work our way up to this, here are a couple of lemmas:

Lemma 2   Suppose is a sequence of r.v.s with characteristic functions . If there exists a r.v. with ch.f. such that

for all then

Lemma 3   Suppose is a r.v. with . Then has the expansion

where .

Summarizing, if ahas zero mean and variance 1,

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). Sequences and Limit Theorems. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec5_4.html. This work is licensed under a Creative Commons License