On the asymptotical behaviour of the maximum in a simple homogeneous Markov chain with large number of states

The paper deals with a sequence of series of trials forming a simple homogeneous Markov chain with transition probabilities
$$ \pi_{ij}=\frac{1}{k}+\frac{\alpha{ij}}{k\varphi(k)}. $$
Here $k$ is the number of states, $\varphi(k)\to\infty$ as $k\to\infty$, $\displaystyle\max_{1\le i,j\le k}|\alpha_{ij}|=O(1)$. Limit distributions of $\displaystyle\rho=\max_{1\le i\le k}h_i$ as $n$ and $k\to\infty$ are investigated, where $h_i$ is the frequency of the $i$th state in $n$ trials.