I have a hard time with the following problem: I consider a population of M r.v. a_i, i.i.d. I need to study the following r.v. eta defined as a function of these r.v.:

(sum a_i^2)^2
eta = ---------------
sum a_i^4

I tried considering the problem by introducing two auxiliary r.v. w_1 and w_2

w_1 = sum a_i^2
w_2 = sum a_i^4

thus defining eta=w_1^2/w_2, whose pdf could be studied, I hope, more easily.

The problem is that w_1 and w_2 are not independent!

Does anyone know how such a problem could be solved or at least approximated? It is important to notice that M is very large, so that w_1 and w_2 could be regarded as converging toward normal distributions, but dependent!

Thanks in advance to anybody