# Dependence of summands of $\sum_{r,c}W_rW_c$

Hi everyone,

I have the following problem and I will be too happy to help me find a solution. Consider a random variable $$K_{ij}= \sum_{r=1}^N\sum_{c=1}^N W_{ir}W_{jc}=\sum_{r=1}^N\sum_{c=1}^N V^{ij}_{rc}$$ where $i \neq j$ and $W \in R$ are i.i.d. random variables $\mathcal{N}(0,1)$, my question is that I can claim $K_{ij}$ is the summation of i.i.d. random variables, I mean we can say that new random variables $V$ s are independent? Thanks a lot in advance

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Obviously not. If X,Y,Z,T are i.i.d normals, then XZ,XT,YZ,YT are dependent (say, because their product is non-negative and each of them is symmetric). I hate to say it, but MO is not primarily designed for making other people do the elementary level work for you (and neither is AoPS). Despite I often disagree with the official MO guidelines, this is a clear case when rereading them would be beneficial... – fedja Jun 2 '11 at 18:49
@fedja, thanks a lot for your answer and nice comment. – Farzad Jun 2 '11 at 20:35