On the waiting times to repeated hits of cells by particles for the polynomial allocation scheme

We consider random polynomial allocations of particles over $N$ cells. Let $ \tau_k, \ k \geq 1, $ be the minimal number of trials when $k$ particles hit the occupied cells. For the case $N\to\infty$ the limit distribution of the random variable $ \tau_k/\sqrt{N} $ is found. An example of application of $\tau_k$ is given.} \keywords{ polynomial allocation, waiting time, occupied cells, distribution density