Limit theorems of probability theory for compact topological groups

We find sufficient conditions for the distribution of the sum of nonidentically distributed elements of compact topological group to converge to the uniform distribution. This conditions generalize the previously known conditions for the weak convergence. We prove that convergence in variation takes place also and that under some additional regularity conditions the uniform convergence to the constant density takes place. For identically distributed summands the rate of convergence is exponential.