Got a bit sidetracked, but here's the problem I'm trying to solve:

Given a set of \( N^4 \) random variables \( \Gamma_{klpq}=(\theta_{kl}+\phi_{pq}) \) mod \( 2\pi \), where \( k,l,p,q \) are integers and \( 1 \le k,l,p,q \le N \).

The following is known:

-Each of the \( N^2 \) random variables \( \theta_{kl} \) and the \( N^2 \) random variables \( \phi_{pq} \) has an *individual* probability distribution which is uniformly distributed over the interval \( [0, 2\pi[ \), and zero otherwise.

-The set of all \( \theta \) and \( \phi \) variables, \( \theta_{11}, \theta_{12},..., \theta_{NN}, \phi_{11}, \phi_{12},..., \phi_{NN} \), are *mutually* independent.

I've managed to show that the set of all \( \Gamma \) variables are *pairwise* independent. That is, any subset of two random \( \Gamma \) variables, \( \Gamma_{klpq} \) and \( \Gamma_{k'l'pq} \) (or, equivalently \( \Gamma_{klpq} \) and \( \Gamma_{klp'q'} \)), are independent. Note that if it wasn't for the modulo \( 2\pi \) in the definition of \( \Gamma_{klpq} \), there would be no pairwise independence.

When looking at more than two random variables it's important to distinguish between *pairwise* independence and *mutual* independence. The latter is a stronger requirement, and allows us to express an expectation of a product as the product of expectations, which is what I need. Therefore, I want to find out whether the set of all \( \Gamma \) variables, \( \Gamma_{1111}, \Gamma_{1112},..., \Gamma_{NNNN} \), are *mutually* independent.