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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 $\{X_n\}$ is a sequence of i.i.d. random variables with mean $ mu < \infty$ and variance $\sigma^2 < \infty$ . Then

\begin{displaymath}\frac{1}{\sqrt{n}} \sum_{i=1}^n (X_i - \mu) \rightarrow X \text{
(in distribution)}
\end{displaymath}

where

\begin{displaymath}X \sim \Nc(0,\sigma^2).
\end{displaymath}

That is,

\begin{displaymath}P\left(\frac{1}{\sqrt{n}} \sum_{i=1}^n (X_i - \mu) \leq x\right)
\rightarrow \int_{-\infty}^x e^{-t^2/2\sigma^2} dt
\end{displaymath}

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 $\{X_n\}$ is a sequence of r.v.s with characteristic functions $\{\phi_n\}$ . If there exists a r.v. $X$ with ch.f. $\phi$ such that

\begin{displaymath}\lim_{n\rightarrow \infty} \phi_n(u) = \phi(u)
\end{displaymath}

for all $u \in \Rbb$ then

\begin{displaymath}X_n \rightarrow X \text{ (in distribution)}.
\end{displaymath}

Lemma 3   Suppose $X$ is a r.v. with $E[X^2] < \infty$ . Then $\phi_X$ has the expansion

\begin{displaymath}\phi_X(u) = 1 + iu E[X] - \frac{u^2}{2}(E[X^2] + \delta(u))
\end{displaymath}

where $\lim_{u\rightarrow 0} \delta(u) = 0$ .


\begin{proof}
of the Central Limit Theorem. For convenience (w.o.l.o.g.), take
...
...he form of a characteristic function of a Gaussian (with zero mean).
\end{proof}

Summarizing, if $X_k$ ahas zero mean and variance 1,

\begin{displaymath}\frac{1}{n}\sum_{k=1}^n X_k \rightarrow 0 \text{ (a.s.)} \end{displaymath}


\begin{displaymath}\frac{1}{\sqrt{n}}\sum_{k=1}^n X_k \rightarrow \Nc(0,1) \text{ (in
distribution)}
\end{displaymath}

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 Creative Commons License