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2020 On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I
Dariusz Buraczewski, Bohdan Dovgay, Alexander Iksanov
Electron. J. Probab. 25: 1-24 (2020). DOI: 10.1214/20-EJP534

Abstract

Consider a weighted branching process generated by the lengths of intervals obtained by stick-breaking of unit length (a.k.a. the residual allocation model) and associate with each weight a ‘box’. Given the weights ‘balls’ are thrown independently into the boxes of the first generation with probability of hitting a box being equal to its weight. Each ball located in a box of the $j$th generation, independently of the others, hits a daughter box in the $(j+1)$th generation with probability being equal the ratio of the daughter weight and the mother weight. This is what we call nested occupancy scheme in random environment. Restricting attention to a particular generation one obtains the classical Karlin occupancy scheme in random environment.

Assuming that the stick-breaking factor has a uniform distribution on $[0,1]$ and that the number of balls is $n$ we investigate occupancy of intermediate generations, that is, those with indices $\lfloor j_{n} u\rfloor $ for $u>0$, where $j_{n}$ diverges to infinity at a sublogarithmic rate as $n$ becomes large. Denote by $K_{n}(j)$ the number of occupied (ever hit) boxes in the $j$th generation. It is shown that the finite-dimensional distributions of the process $(K_{n}(\lfloor j_{n} u\rfloor ))_{u>0}$, properly normalized and centered, converge weakly to those of an integral functional of a Brownian motion. The case of a more general stick-breaking is also analyzed.

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Dariusz Buraczewski. Bohdan Dovgay. Alexander Iksanov. "On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I." Electron. J. Probab. 25 1 - 24, 2020. https://doi.org/10.1214/20-EJP534

Information

Received: 30 May 2020; Accepted: 3 October 2020; Published: 2020
First available in Project Euclid: 8 October 2020

MathSciNet: MR4161133
Digital Object Identifier: 10.1214/20-EJP534

Subjects:
Primary: 60F05, 60J80
Secondary: 60C05

JOURNAL ARTICLE
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