Records in the Infinite Occupancy Scheme

TL;DR

This paper analyzes the distribution of record events in the infinite occupancy scheme, showing that under certain conditions, the scaled process of record times converges to a Poisson process, with joint convergence under regular variation.

Contribution

It establishes the asymptotic Poisson nature of record times in the infinite occupancy scheme and extends results to joint convergence under regular variation conditions.

Findings

Record times form an approximately Poisson process after many balls are thrown.

Joint convergence of multiple record processes is achieved under regular variation.

Results apply when the probability sequence does not decay too quickly.

Abstract

We consider the classic infinite occupancy scheme, where balls are thrown in boxes independently, with probability $p_j$ of hitting box $j$. Each time a box receives its first ball we speak of a record and, more generally, call an $r$-record every event when a box receives its $r$th ball. Assuming that the sequence $(p_j)$ is not decaying too fast, we show that after many balls have been thrown, the suitably scaled point process of $r$-record times is approximately Poisson. The joint convergence of $r$-record processes is argued under a condition of regular variation.