Geodesic
One-sided versions of strong limit theorems
Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 1, pp. 45-61

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Let $\{S_n\}$ be a random walk, $T_n=S_n-\operatorname{med}(S_n)$, $1/2\beta\infty$. Necessary and sufficient conditions for \begin{gather*} \limsup T_n/n^\beta=0\quad \text{a.\,s.} \\ \limsup T_n/n^\beta=1\quad \text{a.\,s.} \\ \limsup T_n/(2n\log\log n)^{1/2}=1\quad \text{a.\,s.} \end{gather*} are obtained. 
@article{TVP_1983_28_1_a2,
     author = {A. I. Martikaǐnen},
     title = {One-sided versions of strong limit theorems},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {45--61},
     publisher = {mathdoc},
     volume = {28},
     number = {1},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1983_28_1_a2/}
}
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A. I. Martikaǐnen. One-sided versions of strong limit theorems. Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 1, pp. 45-61. http://geodesic.mathdoc.fr/item/TVP_1983_28_1_a2/