Suppose $x\sim \text{Normal}(0,\Sigma)$ with diagonal $\Sigma$.
Is there a closed form solution or a good approximation to the following quantity where $y=x/\|x\|$:
$$f(C)=E[yy'Cyy']$$
Target is to understand the growth rate of $g(k)=\operatorname{Tr}(\underbrace{f\circ f \circ \cdots \circ f}_{k} \circ I)$ in terms of $k$.
For unnormalized $x$, I can use Wick's theorem, and this iteration nicely factors, giving nice formulas for $g(k)$.
Wondering if there's a trick I can use to make it work for $y$.
Motivation: factoring density of $x$ is used in Bordelon paper to give loss curve of SGD. Extending this to $y$ will transfer this analysis to the Kaczmarz method.
