The Bernoulli sieve: an overview

TL;DR

This paper reviews the Bernoulli sieve, a probabilistic model for allocating balls into infinitely many boxes using a multiplicative random process, highlighting key limit theorems and new findings on empty box counts.

Contribution

It provides a comprehensive overview of the Bernoulli sieve and introduces new results on the distribution of empty boxes within the occupancy range.

Findings

Limit theorems for the number of occupied boxes

New results on the count of empty boxes

Insights into the residual allocation model

Abstract

The Bernoulli sieve is a version of the classical balls-in-boxes occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative random walk, also known as the residual allocation model or stick-breaking. We give an overview of the limit theorems concerning the number of boxes occupied by some balls out of the first $n$ balls thrown, and present some new results concerning the number of empty boxes within the occupancy range.