On the rate of convergence of quasigroup convolutions of probability distributions

We consider one of the possible generalizations of sums of independent random variable to the case of operations on a finite set, namely quasigroup “sums” that use quasigroup operations on a given finite set instead of the addition operation. For quasigroup “sums” that contain $n$ independent identically distributed random variables we prove that the rate of convergence of distributions to uniform distribution is exponential in $n$.