Maximal Counts in the Stopped Occupancy Problem

TL;DR

This paper analyzes the asymptotic distribution of the maximum occupancy in a generalized coupon-collector problem, revealing oscillatory behavior and approximating it with a convolution of Gumbel distributions.

Contribution

It introduces a novel approximation of the maximal occupancy distribution using convoluted Gumbel distributions and explores oscillatory phenomena on a logarithmic scale.

Findings

Distribution of maximum occupancy does not converge as boxes grow large.

Approximation by convolution of two Gumbel distributions captures oscillations.

Provides insights into moments and tie probabilities for maximum occupancy.

Abstract

We revisit a version of the classic occupancy scheme, where balls are thrown until almost all boxes receive a given number of balls. Special cases are widely known as coupon-collectors and dixie cup problems. We show that as the number of boxes tends to infinity, the distribution of the maximal occupancy count does not converge, but can be approximated by a convolution of two Gumbel distributions, with the approximating distribution having oscillations close to periodic on a logarithmic scale. We pursue two approaches: one relies on lattice point processes obtained by poissonisation of the number of balls and boxes, and the other employs interpolation of the multiset of occupancy counts to a point process on reals. This way we gain considerable insight in known asymptotics obtained previously by mostly analytic tools. Further results concern the moments of maximal occupancy counts and ties for the maximum.