Geodesic
Expectations of the moments of reaching the level by the range type functionals for Markov chain
Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 4, pp. 789-794

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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. 
@article{TVP_1983_28_4_a18,
     author = {Yu. P. Filonov},
     title = {Expectations of the moments of reaching the level by the range type functionals for {Markov} chain},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {789--794},
     publisher = {mathdoc},
     volume = {28},
     number = {4},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1983_28_4_a18/}
}
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Yu. P. Filonov. Expectations of the moments of reaching the level by the range type functionals for Markov chain. Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 4, pp. 789-794. http://geodesic.mathdoc.fr/item/TVP_1983_28_4_a18/