Decomposable statistics in an inverse occupancy problem

We consider a process of the sequential equiprobable allocation of particles among $N$ cells. We assume that up to the start of the trials for cell number $j$ a level $\nu_j$, $j=1,\dots,N$, was established, where $\nu_1,\dots,\nu_N$ are independent identically distributed integer random variables. We carry out the trials until the moment when $k$ cells appear for the first time and their contents reach or exceed the corresponding levels. We study the decomposable statistics
$$ L_{N,k}=\sum^N_{j=1}g(\eta_j), $$
where $g$ is some function of an integer argument, and $\eta_j$ is the content of the $j$-th cell at the moment when the observations cease. We present a general method that reduces the problem of studying the random variables $L_{N,k}$ to the study of the sums of conditionally independent random variables. Using this approach we succeed in obtaining a sufficiently complete description of a class of limit distributions of decomposable statistics in a scheme of equiprobable allocation as $N\to\infty$ and under various modes of change in the parameter $k$.