Geodesic
A uniform estimate of the convergence rate in a~discrete time renewal theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 2, pp. 399-403

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The paper deals with a problem of estimating the convergence rate in discrete time renewal theory. The problem is reduced to the estimation of the probability distribution of a first-arrival time for a denumerable time-homogeneous Markov chain. This gives conditions sufficient for the uniform convergence. 
@article{TVP_1977_22_2_a16,
     author = {V. V. Kala\v{s}nikov},
     title = {A uniform estimate of the convergence rate in a~discrete time renewal theorem},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {399--403},
     publisher = {mathdoc},
     volume = {22},
     number = {2},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1977_22_2_a16/}
}
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V. V. Kalašnikov. A uniform estimate of the convergence rate in a~discrete time renewal theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 2, pp. 399-403. http://geodesic.mathdoc.fr/item/TVP_1977_22_2_a16/