Why is being jointly WSS important in signal estimation with LMMSE estimator?
Recall that in the derivation of the linear MMSE estimator we force the error term $\hat{s}_k-s_k$ to be orthogonal to the signal $y_k$ i.e. $E[(\hat{s}_k-s_k)y_{k-j}] = 0$ for all $j$. The assumption of joint wide sense stationarity allows one to write $E[s_k y_{k-j}]$ as a function of the lag $j$, i.e. the covariance isn't a function of time $k$, just the lag. As a result, the optimal filter coefficients $h_k$ can be obtained by solving a system of linear equations.