On the number of empty boxes in the Bernoulli sieve I

Abstract

The Bernoulli sieve is the infinite ‘balls-in-boxes’ occupancy scheme with random frequencies , where are independent copies of a random variable W taking values in (0, 1). Assuming that the number of balls equals n, let L n denote the number of empty boxes within the occupancy range. The paper proves that, under a regular variation assumption, L n , properly normalized without centring, weakly converges to a functional of an inverse stable subordinator. Proofs rely upon the observation that (log P k ) 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 L n weakly converges (without normalization) the limiting law is mixed Poisson.