On intermediate levels of a nested occupancy scheme in a random environment generated by stick-breaking II. Issue 7 (3rd October 2022) - Iksanov, Alexander; Marynych, Alexander; Samoilenko, Igor
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.