What is usual independence assumptions on adaptive filters
I think there is an error in your referenced independence assumption.
$w(k)$ should be the update part $\Delta w(k)$ i.e the

$w(k+1)=w(k)+\mu \Delta w(k) =w(k)+\mu \frac{x(k)*e(k)}{c+x(k)^Hx(k)}$ 

The error after convergence is uncorelated and zero mean. 

The two other assumptions are correct. 

If your filter is fast enough you may use it at nonstationary systems as long as you are able to track the variations.