2024/12/04 by Oksana Braganets, Braganets, Oksana, Alexander Iksanov +1
Decision Sciences · #Construction Project Management and Performance #FOS: Mathematics #Forecasting Techniques and Applications #Probabilistic and Robust Engineering Design #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2412.03450
openalex publication_date 2024/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Pk = W1 W2⋅ …⋅ Wk-1(1- Wk) 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 Kn(j) the number of occupied boxes in the jth level provided that there are n balls and call the level j intermediate, if j = jn → ∞ and jn = o((log n)a) as n → ∞ for appropriate a>0. We prove that, for some intermediate levels j, finite-dimensional distributions of the process (Kn(\lfloor jn u\rfloor))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 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.