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

TL;DR

This paper investigates the asymptotic behavior of the number of occupied boxes in the Bernoulli sieve, a probabilistic model with infinitely many boxes, deriving limit theorems using renewal theory without restrictive moment conditions.

Contribution

It extends existing results by removing moment constraints, providing new limit theorems for the number of occupied boxes in the Bernoulli sieve.

Findings

Derived new limit distributions for $K_n$ as $n\to\infty$

Removed moment constraints in renewal theory approach

Covered cases previously left open in literature

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 renewal process, also known as the residual allocation model or stick-breaking. We focus on the number $K_n$ of boxes occupied by at least one of $n$ balls, as $n\to\infty$. A variety of limiting distributions for $K_n$ is derived from the properties of associated perturbed random walks. Refining the approach based on the standard renewal theory we remove a moment constraint to cover the cases left open in previous studies.