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Teor. Veroyatnost. i Primenen., 1968, Volume 13, Issue 2, Pages 266–274 (Mi tvp843)  

This article is cited in 4 scientific papers (total in 4 papers)

An extension of the S. N. Bernstein inequalities to multidimensional distributions

Yu. V. Prokhorov

V. A. Steklov Mathematical Institute, USSR Academy of Sciences

Abstract: Let $X_1,…,X_n,…$ be a sequence of identically distributed independent random vectors in $R^m$ and
$$ Y_n=\frac{X_1+…+X_n}{\sqrt n}, $$
Ir$\mathbf EX_j=0$, $|X_j|\le L$ and $n\ge m$, then
$$ \mathbf P\{|Y_n|\ge r\}\le Ce^{-\frac{kr^2}{L^2}} $$
where
$$ c\le1+\frac{e^{5/12}}{\pi/\sqrt2},\quad k\ge\frac1{8e^2}. $$


Full text: PDF file (419 kB)

English version:
Theory of Probability and its Applications, 1968, 13:2, 260–267

Bibliographic databases:

Received: 30.01.1968

Citation: Yu. V. Prokhorov, “An extension of the S. N. Bernstein inequalities to multidimensional distributions”, Teor. Veroyatnost. i Primenen., 13:2 (1968), 266–274; Theory Probab. Appl., 13:2 (1968), 260–267

Citation in format AMSBIB
\Bibitem{Pro68}
\by Yu.~V.~Prokhorov
\paper An extension of the S.\,N.~Bernstein inequalities to multidimensional distributions
\jour Teor. Veroyatnost. i Primenen.
\yr 1968
\vol 13
\issue 2
\pages 266--274
\mathnet{http://mi.mathnet.ru/tvp843}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=230353}
\zmath{https://zbmath.org/?q=an:0167.46903|0165.20301}
\transl
\jour Theory Probab. Appl.
\yr 1968
\vol 13
\issue 2
\pages 260--267
\crossref{https://doi.org/10.1137/1113029}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. F. Götze, Yu. V. Prokhorov, V. V. Ulyanov, “Bounds for characteristic functions of polynomials in asymptotically normal random variables”, Russian Math. Surveys, 51:2 (1996), 181–204  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    2. Theory Probab. Appl., 54:3 (2010), 505–515  mathnet  crossref  crossref  mathscinet  isi
    3. K. N. Pankov, “Otsenki skorosti skhodimosti v predelnykh teoremakh dlya sovmestnykh raspredelenii chasti kharakteristik sluchainykh dvoichnykh otobrazhenii”, PDM, 2012, no. 4(18), 14–30  mathnet
    4. O. V. Denisov, “Statisticheskaya otsenka mnozhestva suschestvennykh argumentov dvoichnoi vektor-funktsii s iskazhennymi znacheniyami”, Matem. vopr. kriptogr., 5:4 (2014), 41–61  mathnet  crossref
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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