Convergence rates of limit theorems in random chord diagrams

Abstract

A chord diagram is a set of n chords joining distinct endpoints on a circle. We study the asymptotic distributions of the number of crossings and the number of simple chords in a random chord diagram. Using size-bias coupling and Stein's method, we obtain bounds on the Wasserstein distance between the distribution of the number of crossings and a standard normal. We also give upper and lower bounds of the same order on the total variation distance between the distribution of the number of simple chords and a Poisson random variable.

Fixed points, descents, and inversions in parabolic double cosets of the symmetric group

“…We remark that Stein's method via size-bias coupling has been successfully used to prove Poisson limit theorems; some examples are: Angel, van der Hofstad, and Holmgren [1] for the number of self-loops and multiple edges in the configuration model, Arratia and DeSalvo [2] for completely effective error bounds on Stirling numbers of the first and second kinds, Goldstein and Reinert [19] for Poisson subset numbers, Holmgren and Janson for sums of functions of fringe subtrees of random binary search trees and random recursive trees, and Paguyo [34] for the number of simple chords in a random chord diagram.…”