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Short Communications
Central limit theorem and the law of large numbers in the mean
V. M. Kruglov Moscow
Abstract:
Let $\{\xi_{n1},\xi_{n2},…,\xi_{nk_n}\}_{n=1}^{\infty}$ be a sequence of independent (for every $n\ge 1$) infinitesimal random variables. We prove that
$$
\lim_{n\to\infty}\mathbf P(\sum_{j=1}^{k_n}\xi_{nj}-A_n<x)=
(2\pi)^{-1/2}\int_{-\infty}^x e^{-u^2/2} du
$$
for some constants $A_n$, $n=1,2,…$, and
$$
\lim_{n\to\infty}\mathbf M|\sum_{j=1}^{k_n}\xi_{nj}-A_n|^{2q}=
(2\pi)^{-1/2}\int_{-\infty}^{\infty}|u|^{2q} e^{-u^2/2} du
$$
for some $q>0$ if and only if
$$
\lim_{n\to\infty}\mathbf M|\sum_{j=1}^{k_n}(\xi_{nj}-
\mathbf M\{\xi_{nj}||\xi_{nj}|<1\})^2-1|^q=0.
$$
Full text:
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English version:
Theory of Probability and its Applications, 1982, 26:4, 813–815
Bibliographic databases:
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
Citation in format AMSBIB
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\by V.~M.~Kruglov
\paper Central limit theorem and the law of large numbers in the mean
\jour Teor. Veroyatnost. i Primenen.
\yr 1981
\vol 26
\issue 4
\pages 824--827
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=636777}
\zmath{https://zbmath.org/?q=an:0488.60046|0474.60026}
\transl
\jour Theory Probab. Appl.
\yr 1982
\vol 26
\issue 4
\pages 813--815
\crossref{https://doi.org/10.1137/1126088}
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http://mi.mathnet.ru/eng/tvp3512 http://mi.mathnet.ru/eng/tvp/v26/i4/p824
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