

\subsection{\bf Proof of Central Limit Theorem}
\subsection{\bf Laplace---I}

\begin{slide}
\begin{itemize}
\item Now we will prove the first (simple) version of the central limit
theorem.

\item From our discussion of characteristic functions 
\[
\varphi_y (k) = \left\lbrack \varphi_x \left(\frac{k}{n}\right) 
\right\rbrack^n
\]
where $\varphi_y$ and $\varphi_x$ are characteristic functions of $y$
and $x$ respectively.

\item Now,
\begin{eqnarray*}
\varphi_x (k) & = & \exp (ik\mu) \int\limits dx e^{ik(x-\mu)} p(x) \\
& = & \exp (ik\mu) \left\lbrack 1 - \frac{k^2 \sigma^2}{2} + O\left( k^3
\right) \right\rbrack
\end{eqnarray*}
\end{itemize}
\end{slide}

