I am now thinking a problem:
what is the expected value of doubly stochastic matrix with i.i.d entries
Each entries is i.i.d in $[0,1]$.
Will the answer be a matrix with all entries $\frac{1}{n}$?
On the other hand, if I view this matrix as a stochastic matrix $A(k)$ and I only care about one realization. Will the time average of this realization be a matrix with all entries $\frac{1}{n}$?
(If both answers are the same, then I could say it is ergodicity in mean.)