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Teor. Veroyatnost. i Primenen., 1982, Volume 27, Issue 3, Pages 566–568 (Mi tvp2388)  

Short Communications

On the non-uniform estimate of the rate of convergence in the local limit theorem in the case of a stable limit distribution

A. S. Mal'kova, V. V. Ul'yanovb

a Leningrad
b Moscow

Abstract: Let $p_n(x)$ be a probability density function of normalized and centered sum of $n$ i. i. d. random variables belonging to the domain of attraction of the stable distribution $G$ of index $\alpha$, $0<\alpha\le 2$, $\alpha\ne 1$. Let $p(x)$ be a probability density function of $G$. It is proved that under certain conditions the relation
$$ \lim_{n\to\infty}|x|^\delta|p_n(x)-p(x)|=0,\qquad 0\le\delta<\alpha\ne 1, $$
holds uniformly in $x$, $-\infty<x<\infty$.

Full text: PDF file (218 kB)

English version:
Theory of Probability and its Applications, 1983, 27:3, 607–609

Bibliographic databases:

Received: 18.09.1980

Citation: A. S. Mal'kov, V. V. Ul'yanov, “On the non-uniform estimate of the rate of convergence in the local limit theorem in the case of a stable limit distribution”, Teor. Veroyatnost. i Primenen., 27:3 (1982), 566–568; Theory Probab. Appl., 27:3 (1983), 607–609

Citation in format AMSBIB
\Bibitem{MalUly82}
\by A.~S.~Mal'kov, V.~V.~Ul'yanov
\paper On the non-uniform estimate of the rate of convergence in the local limit theorem in the case of a~stable limit distribution
\jour Teor. Veroyatnost. i Primenen.
\yr 1982
\vol 27
\issue 3
\pages 566--568
\mathnet{http://mi.mathnet.ru/tvp2388}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=673928}
\zmath{https://zbmath.org/?q=an:0517.60025|0491.60023}
\transl
\jour Theory Probab. Appl.
\yr 1983
\vol 27
\issue 3
\pages 607--609
\crossref{https://doi.org/10.1137/1127067}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1983RJ51700014}


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