Geodesic
A remark on the ergodic theorem for Markov chains
Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 1, pp. 175-177
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An example of an aperiodic transient renewal sequence $\{p_n\}$ with the property $\sum|p_n-p_{n-1}|=\infty$ is constructed. 
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     author = {I. A. Ibragimov},
     title = {A~remark on the ergodic theorem for {Markov} chains},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {175--177},
     year = {1975},
     volume = {20},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1975_20_1_a17/}
}
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I. A. Ibragimov. A remark on the ergodic theorem for Markov chains. Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 1, pp. 175-177. http://geodesic.mathdoc.fr/item/TVP_1975_20_1_a17/