Let's imagine that we have some group $G$ and a number of random variables $X_1, ..., X_n$ on $G$.
We want to consider the function $f(X_1, \ldots , X_n)$ of these r.v.s and to answer some questions about the distribution of this function. Which methods from classical r.v.s could help us? Is there anything like "characteristic function" or other sort of generating functions which could help us to study the distribution?
For concrete example let us consider the group $G = \mathbb{Z}_n$ and random permutations $\pi_1, \ldots, \pi_n$ on $G$, that is, elements of $S_n$ drawn according to uniform distribution.
I'm asking what is the statistical distance between these two random variables: $\pi_1 + \ldots + \pi_n$ and $\rho$, where $\rho$ is random function, that is, a random element of the set of all functions $\mathbb{Z}_n \to \mathbb{Z}_n$ drawn according to uniform distribution.
For one permutation the answer is known (birthday bound), for many permutations there exists some tricks, but I didn't see some really deep theory behind it.