Wating time and testing hypotheses in a multinomial scheme

There are two hypotheses about probabilities $p_1,\dots,p_N$ in a multinomial scheme. These hypotheses are approaching each other as $N$ increases. To distinguish between them, statistical tests based on $\nu_m(k)$ are considered, where $\nu_m(k)$ is the number of trials after which $k$ cells will contain for the first time more than $m$ particles each.