On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking: the case of heavy tails

Abstract

We investigate a nested balls-in-boxes scheme in a random environment. The boxes follow a nested hierarchy, with infinitely many boxes in each level, and the hitting probabilities of boxes are random and obtained by iterated fragmentation of a unit mass. The hitting probabilities of the first-level boxes are given by a stick-breaking model $P_k = W_1 W_2\cdot \ldots\cdot W_{k-1}(1- W_k)$ for $k \in \mathbb{N}$, where $W_1$, $W_2,\ldots$ are independent copies of a random variable $W$ taking values in $(0,1)$. The infinite balls-in-boxes scheme in the first level is known as a Bernoulli sieve. We assume that the mean of $|\log W|$ is infinite and the distribution tail of $|\log W|$ is regularly varying at $\infty$. Denote by $K_n(j)$ the number of occupied boxes in the $j$th level provided that there are $n$ balls and call the level $j$ intermediate, if $j = j_n \to \infty$ and $j_n = o((\log n)^a)$ as $n \to \infty$ for appropriate $a>0$. We prove that, for some intermediate levels $j$, finite-dimensional distributions of the process $(K_n(\lfloor j_n u\rfloor))_{u>0}$, properly normalized, converge weakly as $n\to\infty$ to those of a pathwise Lebesgue-Stieltjes integral, with the integrand being an exponential function and the integrator being an inverse stable subordinator. The present paper continues the line of investigation initiated in the articles Buraczewski, Dovgay and Iksanov (2020) and Iksanov, Marynych and Samoilenko (2022) in which the random variable $|\log W|$ has a finite second moment, and Iksanov, Marynych and Rashytov (2022) in which $|\log W|$ has a finite mean and an infinite second moment.