Asymptotic behavior of some statistics in Ewens random permutations

TL;DR

This paper introduces a general method to determine the limiting distributions of various renormalized statistics in Ewens random permutations, extending results beyond uniform permutations using the method of moments.

Contribution

It presents a novel approach to analyze asymptotic laws for a broad class of statistics in Ewens permutations, including pattern occurrences.

Findings

Asymptotic laws for many permutation statistics are derived.

The method applies to the number of occurrences of any dashed pattern.

Almost independence of events involving images of different integers is established.

Abstract

The purpose of this article is to present a general method to find limiting laws for some renormalized statistics on random permutations. The model considered here is Ewens sampling model, which generalizes uniform random permutations. We describe the asymptotic behavior of a large family of statistics, including the number of occurrences of any given dashed pattern. Our approach is based on the method of moments and relies on the following intuition: two events involving the images of different integers are almost independent.