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Teoriya Veroyatnostei i ee Primeneniya, 1976, Volume 21, Issue 2, Pages 383–387
(Mi tvp3354)
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This article is cited in 14 scientific papers (total in 14 papers)
Short Communications
Relatively stable walks
B. A. Rogosin Novosibirsk
Abstract:
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) $\displaystyle\nu(x)=\int_{\{|\xi_1|<x\}}\xi_1d\mathbf P>0$ for sufficiently large $x>0$, $\displaystyle\qquad\lim_{x\to\infty}\nu(xy)/\nu(x)=1$ for all $y>0\quad$ and
$\displaystyle\quad\lim_{x\to\infty}x\mathbf P(|\xi_1|\ge x)/\nu(x)=0$.
Received: 17.03.1975
Citation:
B. A. Rogosin, “Relatively stable walks”, Teor. Veroyatnost. i Primenen., 21:2 (1976), 383–387; Theory Probab. Appl., 21:2 (1977), 375–379
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https://www.mathnet.ru/eng/tvp3354 https://www.mathnet.ru/eng/tvp/v21/i2/p383
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