  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
$j$th generation, independently of the others, hits a daughter box in the
$(j+1)$th generation with probability being equal the ratio of the 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 $\lfloor j_n u\rfloor$ 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 $j$th
generation. It is shown that the finite-dimensional distributions of the
process $(K_n(\lfloor j_n u\rfloor))_{u>0}$, properly normalized and centered,
converge weakly to those of an integral functional of a Brownian motion. The
case of a more general stick-breaking is also analyzed.
