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Suppose $x_i\in \mathbb{R}^d$ are IID isotropic random vectors with $\|x_i\|=1$ and matrix $A_d$ is defined as follows:

$$A_d=\prod_i^d \left(I-x_ix_i^T\right)$$

Is anything known about the spectrum of $A_d$ for large $d$, like the limiting spectral density? In particular, I'm interested in singular value density around $1$ which appears to be same as singular value density of $X$ around 0 where $X$ is an IID Gaussian matrix, why?

This operator represents $d$ steps of LMS filter on random normalized data.

$k$th moment $\operatorname{Tr}(A^k)\approx d \left(1 - \frac{1}{d}\right)^d \frac{1}{\sqrt{k}}$

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    $\begingroup$ related: mathoverflow.net/q/316603/11260 $\endgroup$
    – Carlo Beenakker
    Commented Jul 21 at 12:15
  • $\begingroup$ @CarloBeenakker thanks, that's indeed related (Kaczmarz). A heuristic argument gives $\operatorname{Tr}(A)$ since multiplying by random projection matrix reduces the expected trace by a factor of $(1-1/d)$ ...curious why $\operatorname{Tr}(A^k)$ decays as $O(1/\sqrt{k})$ $\endgroup$
    – Yaroslav Bulatov
    Commented Jul 21 at 15:37

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