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Teor. Veroyatnost. i Primenen., 1981, Volume 26, Issue 4, Pages 824–827 (Mi tvp3512)  

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
$$ \lim_{n\to\infty}\mathbf M|\sum_{j=1}^{k_n}(\xi_{nj}- \mathbf M\{\xi_{nj}||\xi_{nj}|<1\})^2-1|^q=0. $$


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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
\Bibitem{Kru81}
\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
\mathnet{http://mi.mathnet.ru/tvp3512}
\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}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1982PM42700014}


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