Decomposable statistics and stopping times in a scheme of sampling without replacement

In the framework of the generalized urn scheme, we consider the sampling without replacement. From an urn containing $n$ balls of each of $N$ colours, balls are drawn consequently with equal probabilities and independently of each other. Each ball drawn is not returned to the urn. The sampling of balls is stopped if the numbers of balls of any $k$ different colours drawn from the urn exceed for the first time the levels chosen before the sampling begins. We investigate the statistics of the form
$$ L_{Nk}=\sum_{j=1}^N g(\eta_j), $$
where $g$ is a function of an integer-value argument and $\eta_j$ is the number of balls of the $j$th colour drawn before the sampling is stopped. We obtain a complete description of the class of limit distributions of the statistics $L_{Nk}$ as $N\to\infty$, $n=n(N)$ and $k=k(N)$.