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 ?