On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I

Abstract

Consider a weighted branching process generated by the lengths of intervals obtained by stick-breaking of unit length (a.k.a. the residual allocation model) and associate with each weight a 'box'. Given the weights 'balls' are thrown independently into the boxes of the first generation with probability of hitting a box being equal to its weight. Each ball located in a box of the jth generation, independently of the others, hits a daughter box in the (j + 1)th generation with probability being equal the ratio of the daughter weight and the mother weight. This is what we call nested occupancy scheme in random environment. Restricting attention to a particular generation one obtains the classical Karlin occupancy scheme in random environment.Assuming that the stick-breaking factor has a uniform distribution on [0, 1] and that the number of balls is n we investigate occupancy of intermediate generations, that is, those with indices jnu for u > 0, where jn diverges to infinity at a sublogarithmic rate as n becomes large. Denote by Kn(j) the number of occupied (ever hit) boxes in the jth generation. It is shown that the finite-dimensional distributions of the process (Kn( jnu ))u>0, properly normalized and centered, converge weakly to those of an integral functional of a Brownian motion. The case of a more general stick-breaking is also analyzed.

On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking II

“…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 centered, 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 [4] in which the occupancy of intermediate levels j n → ∞, j n = o((log n) 1/3 ) was analyzed.…”

Limit theorems for discounted convergent perpetuities

“…Such a process has appeared in the recent articles [8], [17] and [18]. The latter paper provides additional references.…”

Limit theorems for discounted convergent perpetuities

“…We have used (20) and (35) for the inequality and the property (a) of B and its consequences (39) for the equality. Invoking (8) we obtain, for large n and appropriate constant C > 0,…”