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 = W₁W₂ ˇ … ˇ W_{k - 1}(1 - W_k) for k ∈ N, where W₁, W₂, … 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 ∞ . Denote by K_n(j) the number of occupied boxes in the jth level provided that there are n balls and call the level j intermediate if j = j_n → ∞ and j_n = o((log n)^a) as n → ∞ for some a > 0. We prove that, for some intermediate levels j, finite-dimensional distributions of the process (K_n(|j_nu|))_{u > 0}, properly normalized, converge weakly as n → ∞ 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 of Buraczewski, Dovgay and Iksanov (2020) and Iksanov, Marynych and Samoilenko (2022) in which the random variable | log W| has a finite second moment, and of Iksanov, Marynych and Rashytov (2022) in which | log W| has a finite mean and an infinite second moment.

2010 AMS Mathematics Subject Classification: Primary 60F05; Secondary 60J80.

Keywords and phrases: Bernoulli sieve, infinite occupancy scheme, perturbed random walk, random environment, weak convergence of finite-dimensional distributions, weighted branching process.