Weak convergence of finite-dimensional distributions of the number of empty boxes in the Bernoulli sieve

TL;DR

This paper studies the asymptotic behavior of the number of empty intervals in the Bernoulli sieve, a random allocation model, showing weak convergence of finite-dimensional distributions as the number of points grows large.

Contribution

It introduces a new approach that relaxes previous restrictions on the distribution of the multiplicative factors in the Bernoulli sieve, advancing theoretical understanding.

Findings

Weak convergence of finite-dimensional distributions established

Relaxed conditions on the multiplicative random walk factors

Enhanced theoretical framework for the Bernoulli sieve

Abstract

The Bernoulli sieve is a random allocation scheme obtained by placing independent points with the uniform [0,1] law into the intervals made up by successive positions of a multiplicative random walk with factors taking values in the interval (0,1). Assuming that the number of points is equal to n we investigate the weak convergence, as n tends to infinity, of finite-dimensional distributions of the number of empty intervals within the occupancy range. A new argument enables us to relax the constraints imposed in previous papers on the distribution of the factor of the multiplicative random walk.