

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

\begin{slide}
\begin{itemize}
\item Then, the new Central Limit Theorem states that the pair ($F,G$)
is asymptotically a (two-dimensional) Gaussian distribution with
probability distribution
\[
p(F,G) \propto \exp \left\lbrace - \frac{n}{2} (y-\mu)^T B(y-\mu)
\right\rbrace
\]

\item where we have lapsed into a vector notation:\ \ $ y = \left\lbrack
\begin{array}{l}
F \\ G
\end{array}
\right\rbrack\,, \hspace{1em} \mu = \left\lbrack
\begin{array}{l}
\mu_f \\ \mu_g
\end{array}
\right\rbrack$ and matrix $B^{-1} = M$.

\item {\bf Proof:}\ \ Just as in ordinary central limit theorem, except
you consider two-dimensional characteristic function $\varphi \left( k_1,
k_2 \right)$.

\item This extension is important as it allows you to use correlations
between functions calculated from the same data (observations).
\end{itemize}
\end{slide}

