I have been struggling for a while with the following problem. Consider a sequence of iid row stochastic matrices $\left\{W_t\right\}_{t=1}^n$ and the linear dynamical system $X(t)=W_tX(t-1)$ with some initial condition $X(0)=\{X_1(0),\dots,X_n(0)\} \in \mathbb{R}^n$, where $X_i(0)$ are iid $\mathcal{N}(0,1)$ r.v.
I am interested in showing that the following expression is increasing in time (or if it does not, at least how it behaves): $\sum\limits_{i,j=1}^n \left\{E\left[\frac{(X_i^t-\bar{X^t})(X_j^t-\bar{X^t})}{\sum\limits_{i=1}^n (X_i^t-\bar{X^t})^2}\right]^2\right\}$.
The expectation is at time $t-1$ and regarding $W^t$.
Any help would be more than welcome.