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Teoriya Veroyatnostei i ee Primeneniya, 1981, Volume 26, Issue 4, Pages 824–827
(Mi tvp3512)
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Short Communications
Central limit theorem and the law of large numbers in the mean
V. M. Kruglov Moscow
Abstract:
Let $\{\xi_{n1},\xi_{n2},\dots,\xi_{nk_n}\}_{n=1}^{\infty}$ be a sequence of independent (for every $n\ge 1$) infinitesimal random variables. We prove that
$$
\lim_{n\to\infty}\mathbf P\biggl(\sum_{j=1}^{k_n}\xi_{nj}-A_n<x\biggr)=
(2\pi)^{-1/2}\int_{-\infty}^x e^{-u^2/2}\,du
$$
for some constants $A_n$, $n=1,2,\dots$, and
$$
\lim_{n\to\infty}\mathbf M\biggl|\sum_{j=1}^{k_n}\xi_{nj}-A_n\biggr|^{2q}=
(2\pi)^{-1/2}\int_{-\infty}^{\infty}|u|^{2q} e^{-u^2/2}\,du
$$
for some $q>0$ if and only if
$$
\lim_{n\to\infty}\mathbf M\biggl|\sum_{j=1}^{k_n}\biggl(\xi_{nj}-
\mathbf M\biggl\{\xi_{nj}\biggl||\xi_{nj}|<1\biggr\}\biggr)^2-1\biggr|^q=0.
$$
Received: 10.05.1979
Citation:
V. M. Kruglov, “Central limit theorem and the law of large numbers in the mean”, Teor. Veroyatnost. i Primenen., 26:4 (1981), 824–827; Theory Probab. Appl., 26:4 (1982), 813–815
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https://www.mathnet.ru/eng/tvp3512 https://www.mathnet.ru/eng/tvp/v26/i4/p824
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