Geodesic
The behaviour of sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 2, pp. 387-391

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In the paper, a necessary and sufficient condition is given in order that $$ -\infty\varliminf_{n\to\infty}\frac{S_n-mS_n}{a_n}\le\varlimsup\frac{S_n-mS_n}{a_n}\infty, $$ where $\{\xi_n\}$ is a sequence of independent random variables, $S_n=\xi_1+\dots+\xi_n$; $m\xi$ is the median of $\xi$; $\{a_n\}$ is an increasing sequence of positive numbers such that there exists, a sequence of indices $\{m_n\}$ for which $$ 1\le\frac{a_{m_{n+1}}}{a_{m_n}}\le C_2\infty. $$ 
@article{TVP_1974_19_2_a12,
     author = {V. M. Kruglov},
     title = {The behaviour of sums of independent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {387--391},
     publisher = {mathdoc},
     volume = {19},
     number = {2},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1974_19_2_a12/}
}
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V. M. Kruglov. The behaviour of sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 2, pp. 387-391. http://geodesic.mathdoc.fr/item/TVP_1974_19_2_a12/