Asymptotic value distribution of additive functions defined on the symmetric group (Q1024106)

In this paper the author examine the asymptotic value distribution of additive functions defined via the multiplicities of lengths of cycles comprising a random permutation taken from the symmetric group with equal probability. He establishes necessary and sufficient conditions for the weak law of large numbers and for the relative compactness of the sequence of distributions. Considering particular cases, the author demonstrate that long cycles play an exceptional role and that, sometimes, in order to obtain a Poisson limit law, their influence must be negligible. The proofs of the main results are based on the seminal I. Z. Ruzsa's ideas, which was used to study classical arithmetic functions.