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.

Keywords:

• Bernoulli sieve
• infinite occupancy scheme
• inverse stable subordinator
• perturbed random walk

Keywords:

• 60F05
• 60K05
• 60C05

Notes

^1. When the law of ξ is d-lattice, the standard form of the key renewal theorem proves (13) with the limit taken over t ∈ dN. Noting that in the case Eξ = ∞ , for any y ∈ R, and using Feller's classical approximation argument (see p. 361–362 in [7]) lead to (13).