On the number of empty boxes in the Bernoulli sieve
Abstract
The Bernoulli sieve is the infinite "balls-in-boxes" occupancy scheme with random frequencies Pk=W1...Wk-1(1-Wk), where (Wk)k∈ are independent copies of a random variable W taking values in (0,1). Assuming that the number of balls equals n, let Ln denote the number of empty boxes within the occupancy range. The paper proves that, under a regular variation assumption, Ln, properly normalized without centering, weakly converges to a functional of an inverse stable subordinator. Proofs rely upon the observation that ( Pk) is a perturbed random walk. In particular, some results for general perturbed random walks are derived. The other result of the paper states that whenever Ln weakly converges (without normalization) the limiting law is mixed Poisson.