On statistics of permutations chosen from the Ewens distribution

TL;DR

This paper investigates the asymptotic behavior of additive functions on permutations under the Ewens distribution, providing conditions for convergence to discrete laws and focusing on Poisson limits, especially for cycle counts.

Contribution

It establishes necessary and sufficient conditions for the weak convergence of additive permutation statistics under Ewens measure, with detailed analysis of cycle count distributions.

Findings

Conditions for convergence to discrete laws are derived.

Poisson distribution limits are characterized for cycle counts.

Results apply to statistics on random permutation matrices.

Abstract

We explore the asymptotic distributions of sequences of integer-valued additive functions defined on the symmetric group endowed with the Ewens probability measure as the order of the group increases. Applying the method of factorial moments, we establish necessary and sufficient conditions for the weak convergence of distributions to discrete laws. More attention is paid to the Poisson limit distribution. The particular case of the number-of-cycles with restricted lengths function is analyzed in more detail. The results can be applied to statistics defined on random permutation matrices.