Expectations of the moments of reaching the level by the range type functionals for Markov chain

Let $S_0,S_1,\dots$ be a homogeneous Markov chain on the set of integers,
$$ \tau_N=\min\{n:\sup_{0\le i,j\le n}(S_i-S_j)\ge N\},\qquad\bar\tau_n=\min\{n:\operatorname{card}\{S_0,\dots,S_n\}=N\}. $$
Theorems on the asymptotical behaviour of $\mathbf M\tau_n$ and $\mathbf M\bar\tau_n$ are proved.