

\subsection{\bf Means---Moments---Correlations}
\subsection{\bf for Multidimensional Distributions II}

\begin{slide}
\begin{itemize}

\item Note that if $\rho_{kl} = 0$---all $k\neq l$---then we say that
$x_1 \ldots x_n$ are uncorrelated.

\item Further, if $x_1 \ldots x_n$ are independent, then Equation~(10)
shows at once that they are uncorrelated.

\item The converse does not follow of course (except under special
circumstances, e.g., if $x_i$ Gaussianly distributed).

\item One can define higher order tensors $M_{klmn\ldots}$ but these
are not too important as in usual applications $x_1 \ldots x_n$ are $n$
functions of $N>>n$ observations.

\begin{description}
\item{$\bullet$} Then central limit theorem---to which we will
come---implies that only the mean and $M_{kl}$ are important
asymptotically (in $N$).
\end{description}
\end{itemize}
\end{slide}

