Geodesic
Limit theorems for the number of occupied boxes in the Bernoulli sieve
Teoriâ slučajnyh processov, Tome 16 (2010) no. 2, pp. 44-57

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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.
Keywords: Infinite occupancy scheme, perturbed random walk, random environment, weak convergence. 
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     author = {Alexander Gnedin and Alexander Iksanov and Alexander Marynych},
     title = {Limit theorems for the number of occupied boxes in the {Bernoulli} sieve},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {44--57},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2010_16_2_a5/}
}
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Alexander Gnedin; Alexander Iksanov; Alexander Marynych. Limit theorems for the number of occupied boxes in the Bernoulli sieve. Teoriâ slučajnyh processov, Tome 16 (2010) no. 2, pp. 44-57. http://geodesic.mathdoc.fr/item/THSP_2010_16_2_a5/