80 research outputs found
On the number of empty boxes in the Bernoulli sieve
The Bernoulli sieve is the infinite "balls-in-boxes" occupancy scheme with
random frequencies , where (W_k)_{k\in\mn} are
independent copies of a random variable taking values in . Assuming
that the number of balls equals , let denote the number of empty boxes
within the occupancy range. The paper proves that, under a regular variation
assumption, , properly normalized without centering, weakly converges to a
functional of an inverse stable subordinator. Proofs rely upon the observation
that 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 weakly converges (without normalization) the
limiting law is mixed Poisson.Comment: Minor corrections to Proposition 5.1 were adde
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