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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
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