On intermediate levels of a nested occupancy scheme in a random environment generated by stick-breaking II. Issue 7 (3rd October 2022)

• Record Type: Journal Article
• Authors: Iksanov, Alexander; Marynych, Alexander; Samoilenko, Igor
• Is Part Of: Stochastics. Volume 94:Issue 7(2022)
• Page Start: 1077
• Page End: 1101
• Publication Date: 2022-10-03
• Subjects: Bernoulli sieve -- GEM distribution -- infinite occupancy -- random environment -- weak convergence -- weighted branching process; Primary: 60F05 -- 60J80 -- Secondary:60C05; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/17442508.2021.2019739
• Languages: English
• ISSNs: 1744-2508

Abstract

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 ( j ) the number of occupied boxes in the j th level and call the level j intermediate if j = j n → ∞ and j n = o ( log ⁡ n ) as n → ∞ . We prove a multidimensional central limit theorem for the vector ( K n ( ⌊ j n u 1 ⌋ ), …, K n ( ⌊ j n u ℓ ⌋ ), properly normalized and centred, as n → ∞, where j n → ∞ and j n = o ( ( 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 → ∞, j n = o ( ( log ⁡ n ) 1 / 3 ) was analysed.