Geodesic
Relatively stable walks
Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 2, pp. 383-387 
B. A. Rogosin. Relatively stable walks. Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 2, pp. 383-387. http://geodesic.mathdoc.fr/item/TVP_1976_21_2_a12/
@article{TVP_1976_21_2_a12,
     author = {B. A. Rogosin},
     title = {Relatively stable walks},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {383--387},
     year = {1976},
     volume = {21},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1976_21_2_a12/}
}
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JO  - Teoriâ veroâtnostej i ee primeneniâ
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IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TVP_1976_21_2_a12/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\xi_1,\xi_2,\dots$ he a sequence of i.i.d.r.v.'s, $\displaystyle S_n=\sum_{k=1}^n\xi_k$, $n=1,\dots$. The following statements are equivalent: 1) $S_n/a_n\to 1$ in probability for some sequence of positive numbers $a_1,a_2,\dots$; 2) $\displaystyle\nu(x)=\int_{\{|\xi_1|0$ for sufficiently large $x>0$, $\displaystyle\qquad\lim_{x\to\infty}\nu(xy)/\nu(x)=1$ for all $y>0\quad$ and $\displaystyle\quad\lim_{x\to\infty}x\mathbf P(|\xi_1|\ge x)/\nu(x)=0$.