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.
Categories: math.PR math.CO