Frank Chang wrote:

> Hi group,
>
> I have a (probably very simple) question that I could not think
> through. i am hoping that you could please enlighten me a little bit.
>
> Suppose I have two completely uncorrelated signals A and B. We can
> assume that they take any form, say, continuous function or discrete
> series. For convenience, let's assume they can be indexed. They have
> the same length N. The question is, does this relationship hold?
>
> <A> * <B> = <A * B>
>
> Here <> represents mean, i.e. <A> = sum(A)/N.
>
> I have done numerical experiments on this, and it appears that it
> hods. I, however, could not come up with an analytical proof. Could
> you please help me here?
>
> A related question is that under what condition this relationship
> breaks down. I know for sure that when A and B are related, the math
> does not hold because <A>^2 != <A^2>. I think an analytical proof for
> the previous question can help with this question as well, but I am
> not sure.

Fun question.

You can always find such i,j,k, so:

sum(A[i]) * sum(B[j]) === sum (A[i]B[i+k])

This holds regardless of A and B.


VLV