

\subsection{\bf Extension of Central Limit Theorem}
\subsection{\bf to Correlated Random Variables---I}

\begin{slide}
\begin{itemize}
\item An important extension of the central limit theorem (necessary
when discussing moments) is

\item Let $x_i$ be independent random variables with the same distribution.

\item Let $f(x)$, $g(x)$ be any function of these random variables.  

\item Suppose we define means, standard deviation and correlations.
\begin{eqnarray*}
\mu_f & = & \langle f\rangle\,, \hspace{2em} \mu_g  =  \langle g\rangle \\
\sigma^2_f & = & \left\langle \left( f - \mu_f \right)^2 \right\rangle \\
\sigma_{fg} & = & \left\langle \left( f - \mu_f \right) \left( g - \mu_g
\right) \right\rangle \\
\sigma^2_g & = & \left\langle \left( g - \mu_g \right)^2 \right\rangle \\
M & = & \left\langle
\begin{array}{ll}
\sigma^2_f & \sigma_{fg} \\
\sigma_{fg} & \sigma^2_g
\end{array}
\right\rangle \hspace{2em} \mbox{moment matrix}
\end{eqnarray*}
\item Finally, set
\begin{eqnarray*}
F & = & \frac{1}{n} \sum^n_{i=1} f\left( x_i \right) \\
G & = & \frac{1}{n} \sum^n_{i=1} g\left( x_i \right) 
\end{eqnarray*}
\end{itemize}
\end{slide}

