12.10 Appendix

LEMMA 12.2   Let $ X, Y$ be standard normal random variables with covariance $ \mathop{\hbox{Cov}}(X,Y) = \rho$, i.e.
$\displaystyle {X \choose Y} \sim \textrm{N}\left( {0 \choose 0},
\left( \begin{array}{cc}
1 & \rho \\
\rho & 1 \\
\end{array} \right)
\right).$     (12.34)

Then we have:

$\displaystyle \mathop{\hbox{Cov}}(X^2, Y^2) = 2 \rho^2 $

PROOF. Define $ Z \sim \textrm{N}(0,1)$ independent of $ X$ and $ X' \stackrel{\mathrm{def}}{=}\rho X + \sqrt{1 - \rho^2} Z$. Then we get:

$\displaystyle {X \choose X'} \sim \textrm{N}\left( {0 \choose 0},
\left( \begin{array}{cc}
1 & \rho \\
\rho & 1 \\
\end{array} \right)
\right).$

$\displaystyle \mathop{\hbox{Cov}}(X^2, Y^2) = \mathop{\hbox{Cov}}(X^2, {X'}^2) = 2 \rho^2 $

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