Über die Wahrscheinlichkeit für nichterscheinen vorgegebener Anzahl der Ausgänge

Suppose $n$ trials are carried out which are connected in a simple homogeneous Markov chain with $k$ outcomes.
Let $P_{n,k}(m)$ be the probability that after $n$ trials exactly $m$ of the $k$ outcomes remain unrealized.
The asymptotic behavior of $P_{n,k}(m)$ is investigated for fixed $m$ for $n\to\infty$ and $k\to\infty$.