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Let $x$ a random Gaussian vector of size $n$ with i.i.d coefficients $N(0,1)$. Let $J$ a random matrix with i.i.d coefficients $N(0,\sigma^2/n)$ where $\sigma \in [0,1]$. For any integer T>n, define: $$R_T=\sum_{k=0}^T (J^k x)(J^k x)'$$ where $A'$ denotes the transpose of A.

The question is : what is the average rank $r(\sigma)$ of $R_T$ as a function of $\sigma$ ?

One may assume that $T$ and/or $n$ are large if it is useful.

My intuition is that $r(\sigma)$ will increase from 0 to $n$ as $\sigma$ increases from $0$ to $1$. Any idea on how to compute this rank ?

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What does this mean? Is $R_T$ a number? If so, what is the rank of a number? – Igor Rivin Nov 14 '13 at 20:39
@Igor: it seems that $R_T$ is a sum of several outer products, so it is a matrix (maybe I'm overseeing something?) – Suvrit Nov 14 '13 at 21:05
Also I think the rank should be should specify how $T$ relates to $n$... – Suvrit Nov 14 '13 at 21:08
Please explain your intuition. Why should the typical rank be less than full? – Brendan McKay Nov 14 '13 at 21:22
@Suvrit ah, I see. That makes some sense... – Igor Rivin Nov 15 '13 at 3:41

Let $\sigma >0,T\geq n-1$. The discriminant of the characteristic polynomial of $J$ is zero with probability $0$ ; then the eigenvalues of $J$ are pairwise distinct and $J$ is a cylic matrix, that implies $x,Jx,\cdots,J^{n-1}x$ is a basis (otherwise $x$ should be an annulator of a polynomial in $J$ of degree $<n$, that is an event with probability $0$). This implies that $R_{n-1}$ has full rank with probability $1$. Certainly, one has the same result when $T\geq n$.

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