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

2010/01/27 by Gnedin, Alexander, Iksanov, Alexander, Marynych, Alexander
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1001.4920

Abstract

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→∞. 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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