Rank of a random matrix

Question

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 ?

Answer 1

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 $T\geq n$.