A functional limit theorem on powers of random permutations

We prove a functional limit theorem for a process defined via partial sums of an additive function on the subset of powers of permutations in the symmetric group. It establishes necessary and sufficient conditions for the convergence to the standard Brownian motion. The main ingredient of the applied approach is estimation of the total-variation distance from the distribution of a cycle structure vector to the distribution of an appropriate random vector with independent coordinates.