

\subsection{\bf Central Limit Theorem}
\subsection{\bf Statement---Liapounoff}

\begin{slide}
\begin{itemize}
\item This extended theorem reads:

\item If $x_1 \ldots x_n$ are independent random variables with
possibly different distributions, and:
\begin{eqnarray*}
\mu_i & = & \left\langle x_i \right\rangle \\
\sigma^2_i & = & \left\langle \left( x_i - \mu_i \right)^2 \right\rangle \\
\rho^3_i & = & \left\langle | x_i - \mu_i |^3 \right\rangle
\end{eqnarray*}

\item where $\rho_i$ {\em must\/} exist (not necessary in Laplace's
version where all $x_i$ had same distribution).

\item Let
\begin{eqnarray*}
\rho^3 & = & \sum_{i=1}^n \rho_i^3 \hspace{2em} \mbox{and} \\
\Sigma^2 & = & \sum_{i=1}^n \sigma_i^2
\end{eqnarray*}

\item Then {\em if\/} $\lim_{n\rightarrow\infty} \rho/\Sigma
\rightarrow 0$,

\item then $y$ is Gaussianly distributed with

\begin{description}
\item{$\bullet$} mean $\mu = \sum^n_{i=1} \mu_i/n$ and
\item{$\bullet$} standard deviation $\Sigma/n$.  
\end{description}

\item Note the $\lim_{n\rightarrow\infty} \rho/\Sigma$ does
$\rightarrow 0$ when $\rho$ exists in Laplace's case as $\rho\propto
n^{1/3}$ and ${\displaystyle \sum}\propto n^{1/2}$.
\end{itemize}
\end{slide}

