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 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.

submitted

References in corpus (2)

• Asymptotic laws for compositions derived from transformed subordinators
• Local limit theorems for finite and infinite urn models

Cited by in corpus (3)

• Lambda-coalescents with dust component
• Weak convergence of finite-dimensional distributions of the number of empty boxes in the Bernoulli sieve
• A law of the iterated logarithm for the number of occupied boxes in the Bernoulli sieve