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

Abstract

A nested occupancy scheme in a random environment is a generalization of the classical Karlin infinite balls-in-boxes occupancy scheme in a 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 $K_n\left(j\right)$ the number of occupied boxes in the jth level and call the level j intermediate if $j=j_n\to \mathrm{\infty }$ and $j_n=o(\log n)$ as $n\to \mathrm{\infty }$. We prove a multidimensional central limit theorem for the vector $\left(K_n\right(\lfloor j_n{u}_1\rfloor ),\dots ,K_n(\lfloor j_n{u}_{\ell }\rfloor )$, properly normalized and centred, as $n\to \mathrm{\infty }$, where $j_n\to \mathrm{\infty }$ and $j_n=o\left(\right(\log n{)}^{1/2})$. The present paper continues the line of investigation initiated in the article [D. Buraczewski, B. Dovgay, and A. Iksanov, On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I, Electron. J. Probab. 25(123) (2020), pp. 1–24] in which the occupancy of intermediate levels $j_n\to \mathrm{\infty }$, $j_n=o\left(\right(\log n{)}^{1/3})$ was analysed.

Keywords:

• Bernoulli sieve
• GEM distribution
• infinite occupancy
• random environment
• weak convergence
• weighted branching process

2010 Mathematics Subject Classifications:

• Primary: 60F05
• 60J80
• Secondary:60C05

Acknowledgments

We thank two anonymous referees for many useful suggestions which significantly improved the presentation of our results.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The present work was supported by the National Research Foundation of Ukraine (project 2020.02/0014 ‘Asymptotic regimes of perturbed random walks: on the edge of modern and classical probability’).