Limit theorems for the number of occupied boxes in the Bernoulli sieve

Abstract

The Bernoulli sieve is a version of the classical `balls-in-boxes' occupancy scheme, in which random frequencies of in¯nitely many boxes are produced by a multiplicative renewal process, also known as the residual allocation model or stick-breaking. We focus on the number Kn of boxes occupied by at least one of n balls, as n ! 1. A variety of limiting distributions for Kn is derived from the properties of associated perturbed random walks. Re¯ning the approach based on the standard renewal theory we remove a moment constraint to cover the cases left open in previous studies.