Limit theorems for the number of occupied boxes in the Bernoulli sieve

Details Limit theorems for the number of occupied boxes in the Bernoulli sieve Limit theorems for the number of occupied boxes in the Bernoulli sieve

2010-01-27

arxiv.org/abs/1001.4920v1

Mathematics

Probability

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Scientific paper

The Bernoulli sieve is a version of the classical `balls-in-boxes' occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative renewal process, also known as the residual allocation model or stick-breaking. We focus on the number $K_n$ of boxes occupied by at least one of $n$ balls, as $n\to\infty$. A variety of limiting distributions for $K_n$ is derived from the properties of associated perturbed random walks. Refining the approach based on the standard renewal theory we remove a moment constraint to cover the cases left open in previous studies.

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