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Zapiski Nauchnykh Seminarov LOMI, 1982, Volume 119, Pages 87–92
(Mi znsl3987)
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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 $\varphi(x)=\sqrt x\quad(\varphi(x)=\sqrt{x\ln\ln x})$.
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
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https://www.mathnet.ru/eng/znsl3987 https://www.mathnet.ru/eng/znsl/v119/p87
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