On a sequential test

Let, in the scheme of independent and identical trials with $N$ outcomes, $\nu_m(k)$ be the number of trials after which $k$ outcomes have been realized for the first time more than $m$ times each. We study $\nu_m(k)$ (as $N\to\infty$) as a random function of parameter $k$ and construct a multidimensional analogue of the statistical test of [1].