Weak Convergence of Finite-Dimensional Distributions of the Number of Empty Boxes in 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 weak convergence, as $n\to\infty$, 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.