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Limit theorems for the number of occupied boxes in the Bernoulli sieve. (English) Zbl 1249.60029
Summary: 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. A refined approach based on the standard renewal theory allows us to remove a moment constraint and to cover the cases left open in previous studies.

##### MSC:
 60F05 Central limit and other weak theorems 60C05 Combinatorial probability
##### Keywords:
Bernoulli sieve; renewal process; perturbed random walk
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