Geodesic
On the rate of convergence in the strong law of large numbers
Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 1, pp. 138-143

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Let $X_1,X_2,\dots$ be independent random variables, $S_n=X_1+\dots+X_n$, $\{b_n\}_{n=1}^\infty$ be a positive nondecreasing sequence, $\{n_i\}_{i=1}^\infty$ be an increasing sequence of integers satisfying some conditions. We obtain relations between $\displaystyle\mathbf P\{\sup_{k\ge n_m}S_k/b_k\ge\varepsilon\}$ and $$ Q_m(\varepsilon)=\mathbf P\{S_{n_m}\ge \varepsilon b_{n_m}\}+\sum_{k=m}^\infty\mathbf P\{S_{n_{k+1}}-S_{n_k}\ge\varepsilon b_{n_{k+1}}\},\qquad\varepsilon>0,m\ge 1. $$ 
@article{TVP_1981_26_1_a10,
     author = {L. V. Rozovskiǐ},
     title = {On the rate of convergence in the strong law of large numbers},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {138--143},
     publisher = {mathdoc},
     volume = {26},
     number = {1},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_1_a10/}
}
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L. V. Rozovskiǐ. On the rate of convergence in the strong law of large numbers. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 1, pp. 138-143. http://geodesic.mathdoc.fr/item/TVP_1981_26_1_a10/