

\subsection{\bf Central Limit Theorem for Quotients}

\begin{slide}
\begin{itemize}
\item Corollary of correlated central limit theorem:\ \ The quotient
$F/G$ is asymptotically Gaussian with mean $\mu_f/\mu_g$ and standard
deviation
\[
\frac{\langle f\rangle}{\langle g\rangle} \sqrt{\frac{1}{n}
\left\langle \left( \frac{f}{\langle f\rangle} - \frac{g}{\langle
g\rangle} \right)^2 \right\rangle}
\]

\item Examples of above result are:
\begin{description}
\item{$\bullet$ (a)} $G=1$, then $F/G$ has mean$=\mu_f$, $\sigma =
\sqrt{\frac{1}{n} \sigma^2_f}$ , i.e., we recover the ordinary central
limit theorem

\item{$\bullet$ (b)} $F=G$, then $F/G$ has mean$=1$, $\sigma=0$.  This
is what one would hope.
\end{description}
\end{itemize}
\end{slide}

