I'm looking for a theorem that states that under certain coditions (which? I'm flexible.), the expected value of a function that works on vectors, will be the same for all orthogonal vectors.

Namely, given a function
$\displaystyle f : R^n \rightarrow R^n$ ( or $\displaystyle f : N^n \rightarrow N^n$ , doesn't matter to me ) ,
and under certain conditions, this holds:
$\displaystyle E[f(x_1,x_2,..,x_n)] = E[f(y_1,y_2,..,y_n)]$, for x $\displaystyle \perp$ y
Is there anything that even remotely suggests this kind of thing?
(I'm talking about large n values, of\course.)

I know this sounds like a statistics problem, I just figured there's more chance that discrete math \ set theory people would use this kind of theorem..