No one answers my questions, but I figure I might as well try.

In solving a problem I've stumbled across something that I'd like to verify. Suppose $\displaystyle X \sim N(0, \sigma^2 I_p) $ and $\displaystyle Q$ is a symmetric $\displaystyle p \times p$ matrix. Then, for any function $\displaystyle f(\cdot)$ the following holds:

$\displaystyle E \left\{f(\|X\|^2) X^T Q X\right\} = E \left\{f(\|X\|^2) \|X\|^2 \right\} \frac{tr(Q)}{p}$

The idea behind this is to apply the spectral theorem to write $\displaystyle \Gamma D \Gamma'$ and define $\displaystyle Y = \Gamma' X$. Then, $\displaystyle Y \stackrel{d}{=} X$ and $\displaystyle \|X\|^2 \equiv \|Y\|^2$.

From there, argue

$\displaystyle E \left\{f(\|X\|^2) X^T Q X\right\}

= E \left\{f(\|Y\|^2) Y^T D Y\right\}

= E \sum_{i = 1} ^ p \lambda_i Y_i ^ 2 f(\|Y\|^2)$

$\displaystyle = \sum_{i = 1} ^ p \lambda_i E [Y_i ^ 2 f(\|Y\|^2)]

= E [Y_i ^ 2 f(\|Y\|^2)] tr(Q)

= \frac{tr(Q)}{p} \sum_{i = 1} ^ p E [Y_i ^ 2 f(\|Y\|^2)]$

$\displaystyle = E \left\{f(\|X\|^2) \|X\|^2 \right\} \frac{tr(Q)}{p}$

(The $\displaystyle \lambda_i$ are the eigenvalues of $\displaystyle Q$ if that isn't clear). I guess the main thing I need to get this off the ground is that $\displaystyle f(\|Y\|^2) Y_i$ are identically distributed for all i. This seems true since I think $\displaystyle (\|Y\|^2, Y_i)$ is identically distributed for all i. This justifies the steps where I do stuff with the expected values.