If you use $(1+x^2)^{-a}=\sum_{n\geq 0} (-1)^n \binom{n+a-1}a x^{2n}$ and then take moments, it seems like you get a wildly oscillating series. And yet your expectations appear to be real numbers that converge to $0$ as $a$ grows.

@Dirk It was not an objection, but a comment. IMHO the question is defective. It is meaningless to talk about randomness when there is no specified distribution.

This is not going to be uniform.

This is generalization to the question mathoverflow.net/questions/73805/…, which is open.

We keep rediscovering this beautiful identity! See math.stackexchange.com/questions/102242/… and mathoverflow.net/questions/110394/… and math.stackexchange.com/questions/5025/…

@BenWebster Do you mean "in all the different cases where $f_i=f$"?