2020/11/23 by Iksanov, Alexander, Marynych, Alexander, Samoilenko, Igor
#60J80 #FOS: Mathematics #Primary: 60F05 #Probability (math.PR) #Secondary: 60C05
paper · doi:10.48550/arxiv.2011.12231
A nested occupancy scheme in random environment is a generalization of the classical Karlin infinite balls-in-boxes occupancy scheme in 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 Kn(j) the number of occupied boxes in the jth level and call the level j intermediate if j=jn→∞ and jn=o(log n) as n→∞. We prove a multidimensional central limit theorem for the vector (Kn(\lfloor jn u1\rfloor),…, Kn(\lfloor jn u_ℓ\rfloor), properly normalized and centered, as n→∞, where jn→∞ and jn=o((log n)1/2). The present paper continues the line of investigation initiated in Buraczewski, Dovgay and Iksanov [Electron. J. Probab. 25: paper no. 123, 2020] in which the occupancy of intermediate levels jn→∞, jn=o((log n)1/3) was analyzed.