Geodesic
On the rate of convergence for countable Markov chains
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 2, pp. 395-399 
N. N. Popov. On the rate of convergence for countable Markov chains. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 2, pp. 395-399. http://geodesic.mathdoc.fr/item/TVP_1979_24_2_a13/
@article{TVP_1979_24_2_a13,
     author = {N. N. Popov},
     title = {On the rate of convergence for countable {Markov} chains},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {395--399},
     year = {1979},
     volume = {24},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_2_a13/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $Z_n$ he an ergodic Markov chain with state space $\Omega=\{0,1,\dots\}$ and $\tau_{ij}=\min\{n\ge 1\colon Z_n=j\ (Z_0=i)\}$. We find necessary and sufficient conditions for $\mathbf M\tau_{ij}^{\gamma}<\infty$ ($\gamma\ge 1$). It is proved that the condition $\mathbf M\tau_{ij}^{\gamma}<\infty$ is sufficient for the existence of $C(k)<\infty$ such that $$ |p_{ij}^{(n)}-\pi_j|\le C(k)n^{1-\gamma}\mathbf M\tau_{ik}^{\gamma},\qquad n=1,2,\dots, $$ where $p_{ij}^{(n)}=\mathbf P\{Z_n=j\mid Z_0=i\}$, $\displaystyle\pi_j=\lim_{n\to\infty}p_{ij}^{(n)}$.