What is usual independence assumptions on adaptive filters In this paper stands:


  The derivation and analysis of NLMS rest upon the usual independence assumptions.


It has a footnote:


  The independence assumptions used in the analysis of adaptive filters
  are:
  
  
  sequences $x(k)$ and $w(k)$ are zero mean, stationary, jointly normal,
  and with finite moments
  the successive increments of tap weights are independent of one
  another; and
  the error and $x(k)$ sequences are statistically independent of one
  another
  


I have several problems with that:


Who and when choose this assumptions? Is it something general or it just appears in the first publication on this topic?
I think in practice it is not possible to work just with stationary inputs $x(k)$. And also input and the adaptive weights also will never be zero mean (especially during online usage when it is impossible to normalize the input data). Also the error and input are strongly correlated, so how it can be statistically independent? 
Is there some other theory dealing with analyzing those more real-life applications where those assumptions cannot be fulfiled?


If is it possible to answer, than please answer. If my questions are too stupid, please correct me. Thanks in advance.
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.