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Consider a weighted branching process generated by the lengths of intervalsobtained by stick-breaking of unit length (a.k.a. the residual allocationmodel) and associate with each weight a `box'. Given the weights `balls' arethrown independently into the boxes of the first generation with probability ofhitting a box being equal to its weight. Each ball located in a box of the$j$th generation, independently of the others, hits a daughter box in the$(j+1)$th generation with probability being equal the ratio of the daughterweight and the mother weight. This is what we call nested occupancy scheme inrandom environment. Restricting attention to a particular generation oneobtains 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 intermediategenerations, that is, those with indices $\lfloor j_n u\rfloor$ 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 $j$thgeneration. It is shown that the finite-dimensional distributions of theprocess $(K_n(\lfloor j_n u\rfloor))_{u>0}$, properly normalized and centered,converge weakly to those of an integral functional of a Brownian motion. Thecase of a more general stick-breaking is also analyzed.

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