Geodesic
An Estimate of the Remainder Term in a~Limit Theorem for Recurrent Events
Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 2, pp. 327-331

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Let $\mathcal{E}$ be a recurrent event, $a_n$ be the probability that $\mathcal{E}$ occurs at the $n$-th trial and $p_n$ be the probability that $\mathcal{E}$ occurs for the first time at the $n$-th trial. A. N. Kolmogorov [2] proved that as $n\to\infty$ $$ B_n=a_n-\frac{1}{\mu}\to 0, $$ where $\mu=\sum_{k\geqq 1}kp_n$ and then W. Feller [3] estimated the remainder term $B_n$ under some addition-conditions. In this note a more exact estimate of $B_n$ under more general conditions as compared to Feller's is given. 
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     author = {A. O. Gel'fond},
     title = {An {Estimate} of the {Remainder} {Term} in {a~Limit} {Theorem} for {Recurrent} {Events}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {327--331},
     publisher = {mathdoc},
     volume = {9},
     number = {2},
     year = {1964},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1964_9_2_a10/}
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A. O. Gel'fond. An Estimate of the Remainder Term in a~Limit Theorem for Recurrent Events. Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 2, pp. 327-331. http://geodesic.mathdoc.fr/item/TVP_1964_9_2_a10/