

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

\begin{slide}
\begin{itemize}
\item So, the characteristic function of $y$ is:
\begin{eqnarray*}
\varphi_y (k) & = & \exp (ik\mu) \left\lbrack 1 - \frac{k^2 \sigma^2}{2n^2} + 
O\left( \frac{k}{n} \right)^3 \right\rbrack^n \\
& = & e^{ik\mu} \left\lbrace \exp \left\lbrack - \frac{k^2 \sigma^2}{2n^2} + 
O\left( \frac{k}{n} \right)^3 \right\rbrack \right\rbrace^n \\
& = & \exp (ik\mu ) \exp \left\lbrace - \frac{k^2 \sigma^2}{2n} + 
O\left( \frac{k^3}{n^2} \right) \right\rbrace
\end{eqnarray*}

\item Now, $\frac{k^3}{n^2} \rightarrow 0$ faster than $\frac{k^2}{n}$
as $n\rightarrow\infty$ and the Fourier transform of a Gaussian, is
well known to also be a Gaussian, 

\item so we find probability distribution:

\[
p(y) = \frac{1}{\sqrt{2\pi}}\ \frac{\sqrt{n}}{\sigma} \exp \left\lbrace -
\frac{n(x-\mu)^2}{2\sigma^2} \right\rbrace
\]

\item and $y$ is Gaussianly distributed as claimed.
\end{itemize}
\end{slide}

