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Izv. Akad. Nauk SSSR Ser. Mat., 1980, Volume 44, Issue 4, Pages 868–885 (Mi izv1858)  

On estimates and the asymptotic behavior of the probability of nonintersection of moving boundaries by sums of independent random variables

A. A. Novikov


Abstract: This paper studies estimates and the asymptotic behavior as $n\to\infty$ for the probabilities $\mathbf P\{|S_k|\leqslant f(k), m\leqslant k\leqslant n\}$ and $\mathbf P\{S_k\geqslant g(k),  m\leqslant k\leqslant n\}$, where $S_n=\sum_{k=1}^n\xi_k$, the $\xi_k$ being independent identically distributed random variables with mean zero, and $f(n)$ and $g(n)$ are nonrandom functions. Under certain restrictions on the boundaries $f(n)$ and $g(n)$ logarithmic asymptotes of these probabilities are found in the case when the $\xi_k$ satisfy (respectively) a two-sided or a one-sided Cramér condition. The method is based on an absolutely continuous substitution for the original probability measure.
Bibliography: 18 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1981, 17:1, 129–145

Bibliographic databases:

UDC: 519.2
MSC: Primary 60G50, 60G40, 60J50; Secondary 41A60
Received: 05.07.1979

Citation: A. A. Novikov, “On estimates and the asymptotic behavior of the probability of nonintersection of moving boundaries by sums of independent random variables”, Izv. Akad. Nauk SSSR Ser. Mat., 44:4 (1980), 868–885; Math. USSR-Izv., 17:1 (1981), 129–145

Citation in format AMSBIB
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\by A.~A.~Novikov
\paper On estimates and the asymptotic behavior of the probability of non\-in\-ter\-section of moving boundaries by sums of independent random variables
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1980
\vol 44
\issue 4
\pages 868--885
\mathnet{http://mi.mathnet.ru/izv1858}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=587341}
\zmath{https://zbmath.org/?q=an:0464.60052|0451.60024}
\transl
\jour Math. USSR-Izv.
\yr 1981
\vol 17
\issue 1
\pages 129--145
\crossref{https://doi.org/10.1070/IM1981v017n01ABEH001324}
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  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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