An Extension of S. N. Bernstein’s Inequalities to Multidimensional Distributions

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Previous article Next article An Extension of S. N. Bernstein’s Inequalities to Multidimensional DistributionsYu. V. ProkhorovYu. V. Prokhorovhttps://doi.org/10.1137/1113029PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. N. Bernstein, On a modification of Chebyshev's inequality and on the error in Laplace's formulaCollected Works, Vol. 4, Izd-vo `Nauka', Moscow, 1964, 71–80, (In Russian.) Google Scholar[2] S. N. Bernstein, Probability Theory, GTTI, Moscow, 1964, (In Russian.) Google Scholar[3] A. Kolmogoroff, Über das Gesetz des iterierten Logarithmus, Math. Ann., 101 (1929), 126–135 10.1007/BF01454828 MR1512520 CrossrefGoogle Scholar[4] Yu. V. Prokhorov, An extremal problem in probability theory, Theor. Probability Appl., 4 (1959), 201–203 10.1137/1104017 MR0121857 (22:12587) 0093.15102 LinkGoogle Scholar[5] S. Bennett, Probability inequalities for a sum of independent random variables, J. Amer. Statist. Assoc., 59 (1962), 33–45 CrossrefGoogle Scholar[6] Wassily Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58 (1963), 13–30 MR0144363 (26:1908) 0127.10602 CrossrefGoogle Scholar[7] A. A. Borovkov, Some inequalities for sums of multidimensional random variables, Theory Prob. Applications, 13 (1968), 156–160 10.1137/1113013 0196.19501 LinkGoogle Scholar[8] Yu. V. Prokhorov, Convergence of stochastic processes and limit theorems in probability theory, Theory Prob. Applications, 1 (1956), 157–214 10.1137/1101016 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Evaluating probabilistic forecasts of extremes using continuous ranked probability score distributionsInternational Journal of Forecasting, Vol. 7 | 1 Aug 2022 Cross Ref BibliographyFundamental Aspects of Operational Risk and Insurance Analytics | 23 January 2015 Cross Ref Sharp Bounds for the Tails of Functionals of Markov ChainsP. Bertail and S. ClémençonTheory of Probability & Its Applications, Vol. 54, No. 3 | 26 August 2010AbstractPDF (214 KB)A large deviation inequality for vector functions on finite reversible Markov ChainsThe Annals of Applied Probability, Vol. 17, No. 4 | 1 Aug 2007 Cross Ref The Exact and Asymptotic Distributions of Cramér-Von Mises StatisticsJournal of the Royal Statistical Society: Series B (Methodological), Vol. 58, No. 1 | 5 December 2018 Cross Ref On Exponential Bounds for Probabilities of Large DeviationsV. V. YurinskiiTheory of Probability & Its Applications, Vol. 37, No. 1 | 28 July 2006PDF (208 KB)Exponential Bounds for the Distribution of the Maximum of a Non-Gaussian Random FieldE. I. OstrovskiiTheory of Probability & Its Applications, Vol. 35, No. 3 | 17 July 2006AbstractPDF (1196 KB)A refinement of S. N. Bernstein's inequalityMathematical Notes of the Academy of Sciences of the USSR, Vol. 47, No. 6 | 1 Jun 1990 Cross Ref The Empirical Characteristic Process When Parameters Are EstimatedContributions to Probability | 1 Jan 1981 Cross Ref Probability inequalities for sums of absolutely regular processes and their applicationsZeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 43, No. 4 | 1 Jan 1978 Cross Ref Summary of Papers Presented at Sessions of the Probability and Statistics Section of the Moscow Mathematical Society (February–October 1974)Theory of Probability & Its Applications, Vol. 20, No. 2 | 17 July 2006AbstractPDF (1740 KB)Multidimensional Limit Theorems for Large DeviationsL. V. OsipovTheory of Probability & Its Applications, Vol. 20, No. 1 | 17 July 2006AbstractPDF (1239 KB)On Convergence in the $L_2$-Norm of Probability Density EstimatesE. A. NadarayaTheory of Probability & Its Applications, Vol. 18, No. 4 | 28 July 2006AbstractPDF (311 KB)On the S. N. Bernstein InequalityS. L. Blyumin and B. D. KotlyarTheory of Probability & Its Applications, Vol. 16, No. 1 | 17 July 2006AbstractPDF (287 KB)On Multidimensional Analogues of S. N. Bernstein’s InequalitiesA. V. ProkhorovTheory of Probability & Its Applications, Vol. 13, No. 2 | 28 July 2006AbstractPDF (931 KB)Some Remarks on Multidimensional Bernstein–Kolmogorov-Type InequalitiesV. M. ZolotarevTheory of Probability & Its Applications, Vol. 13, No. 2 | 28 July 2006AbstractPDF (458 KB)S. N. Bernstein’s Inequalities in the Multidimensional CaseA. V. ProkhorovTheory of Probability & Its Applications, Vol. 13, No. 3 | 17 July 2006AbstractPDF (652 KB) Volume 13, Issue 2| 1968Theory of Probability & Its Applications197-358 History Submitted:30 January 1968Published online:28 July 2006 InformationCopyright © 1968 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1113029Article page range:pp. 260-267ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics