Convergence rates of limit theorems in random chord diagrams

Abstract

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 Kolmogorov distance between the distribution of the number of crossings and a standard normal random variable, and on the total variation distance between the distribution of the number of simple chords and a Poisson random variable. As an application, we provide explicit error bounds on the number of chord diagrams containing no simple chords.

12 pages, 1 figure. Strengthened main theorem to the Kolmogorov distance. Added a new section. Made some additional minor revisions

References in corpus (2)

• A central limit theorem for descents of a Mallows permutation and its inverse
• An Asymptotic Expansion for the Number of 2-Connected Chord Diagrams