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

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 left perpendicularj(n)uright perpendicular for u > 0, where j(n) diverges to infinity at a sublogarithmic rate as n becomes large. Denote by K-n(j) the number of occupied (ever hit) boxes in the jth generation. It is shown that the finite-dimensional distributions of the process (Kn(left perpendicularj(n)uright perpendicular))(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.