On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking II

Abstract

A nested occupancy scheme in random environment is a generalization of the classical Karlin infinite balls-in-boxes occupancy scheme in random environment (with random probabilities). Unlike the Karlin scheme in which the collection of boxes is unique, there is a nested hierarchy of boxes, and the hitting probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. In the present paper we assume that the random fragmentation law is given by stick-breaking in which case the infinite occupancy scheme defined by the first level boxes is known as the Bernoulli sieve. Assuming that $n$ balls have been thrown, denote by $K_n(j)$ the number of occupied boxes in the $j$th level and call the level $j$ intermediate if $j=j_n\to\infty$ and $j_n=o(\log n)$ as $n\to\infty$. We prove a multidimensional central limit theorem for the vector $(K_n(\lfloor j_n u_1\rfloor),\ldots, K_n(\lfloor j_n u_\ell\rfloor)$, properly normalized and centered, as $n\to\infty$, where $j_n\to\infty$ and $j_n=o((\log n)^{1/2})$. The present paper continues the line of investigation initiated in Buraczewski, Dovgay and Iksanov [Electron. J. Probab. 25: paper no. 123, 2020] in which the occupancy of intermediate levels $j_n\to\infty$, $j_n=o((\log n)^{1/3})$ was analyzed.