On the number of empty boxes in the Bernoulli sieve

Details On the number of empty boxes in the Bernoulli sieve On the number of empty boxes in the Bernoulli sieve

2011-04-12

arxiv.org/abs/1104.2299v2

Mathematics

Probability

Minor corrections to Proposition 5.1 were added

Scientific paper

The Bernoulli sieve is the infinite "balls-in-boxes" occupancy scheme with random frequencies $P_k=W_1...W_{k-1}(1-W_k)$, where $(W_k)_{k\in\mn}$ 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 centering, 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.

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