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Zapiski Nauchnykh Seminarov LOMI, 1982, Volume 119, Pages 87–92 (Mi znsl3987)  

On connection berween the law of large numbers for squares and the law of iterated logarithm

V. A. Egorov
Abstract: Let $\{X_i\}$ be a sequence of independent random variables with $EX_i=0, i=1,2,\dots,\quad b_n\uparrow\infty$ be a sequence of real numbers. Under some conditions it is proved that if $\sum_{i=1}^nX_i^2\stackrel{p}{=}O(b_n)$ $(\sum_{i=1}^nX_i^2=O(b_n) \text{ a.s.})$, then $\sum_{i=1}^nX_i\stackrel{p}{=}O(\varphi(b_n))$ $(\sum_{i=1}^nX_i=O(\varphi(b_n)) \text{ a.s.})$, where $\varphi(x)=\sqrt x\quad(\varphi(x)=\sqrt{x\ln\ln x})$.
Bibliographic databases:
Document Type: Article
UDC: 519.21
Language: Russian
Citation: V. A. Egorov, “On connection berween the law of large numbers for squares and the law of iterated logarithm”, Problems of the theory of probability distributions. Part VII, Zap. Nauchn. Sem. LOMI, 119, "Nauka", Leningrad. Otdel., Leningrad, 1982, 87–92
Citation in format AMSBIB
\Bibitem{Ego82}
\by V.~A.~Egorov
\paper On connection berween the law of large numbers for squares and the law of iterated logarithm
\inbook Problems of the theory of probability distributions. Part~VII
\serial Zap. Nauchn. Sem. LOMI
\yr 1982
\vol 119
\pages 87--92
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl3987}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=666088}
\zmath{https://zbmath.org/?q=an:0496.60027}
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