Remarks on Inequalities for Large Deviation Probabilities

Previous article Next article Remarks on Inequalities for Large Deviation ProbabilitiesI. F. Pinelis and A. I. SakhanenkoI. F. Pinelis and A. I. Sakhanenkohttps://doi.org/10.1137/1130013PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. N. Bernstein, Probability Theory, Gostekhizdat, Moscow–Leningrad, 1946, (In Russian.) Google Scholar[2] V. V. Yurinskii, Exponential bounds for large deviations, Theory Prob. Appl., 19 (1974), 154–155 0323.60029 LinkGoogle Scholar[3] V. V. Yurinskii, On an infinite-dimensional version of S. N. Bernstein's inequalities, Theory Prob. Appl., 15 (1970), 108–109 0216.46601 LinkGoogle Scholar[4] S. V. Nagaev and , D. Kh. Fuk, Probability inequalities for sums of independent random variables, Theory Prob. Appl., 16 (1971), 643–550 0259.60024 LinkGoogle Scholar[5] M. X. Fernique, Intégrabilité des vecteurs gaussiens, C. R. Acad. Sci. Paris Sér. A-B, 270 (1970), A1698–A1699 42:1170 0206.19002 Google Scholar[6] S. V. Nagaev, Probabilities of large deviations for a Gaussian distribution in Banach space, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk, (1981), 18–21, 94, (In Russian.) 83c:60008 0478.60016 Google Scholar[7] I. F. Pinelis, Limit theorems on large deviations with Cramér's condition violated for sums of infinite-dimensional random variables, Deposited in the VINITI, No. 1674-81. (In Russian.) Google Scholar[8] A. V. Skorokhod, A note on Gaussian measures in a Banach space, Theory Prob. Appl., 15 (1970), 508– 0218.28008 LinkGoogle Scholar[9] E. Hewitt and , K. Ross, Abstract harmonic analysis. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 115, Springer-Verlag, Berlin, 1979ix+519, 2nd ed., New York 81k:43001 CrossrefGoogle Scholar[10] To Ting On and , Wing Yip Kai, A generalized Jensen's inequality, Pacific J. Math., 58 (1975), 255–259 51:14233 0416.43001 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Improved concentration bounds for sums of independent sub-exponential random variablesStatistics & Probability Letters, Vol. 191 Cross Ref Sample average approximations of strongly convex stochastic programs in Hilbert spaces28 May 2022 | Optimization Letters, Vol. 58 Cross Ref Kernel conjugate gradient methods with random projectionsApplied and Computational Harmonic Analysis, Vol. 55 Cross Ref Multi-scale vector quantization with reconstruction trees12 May 2020 | Information and Inference: A Journal of the IMA, Vol. 10, No. 3 Cross Ref Multi-scale vector quantization with reconstruction trees11 May 2020 | Information and Inference: A Journal of the IMA, Vol. 32 Cross Ref Optimal rates for spectral algorithms with least-squares regression over Hilbert spacesApplied and Computational Harmonic Analysis, Vol. 48, No. 3 Cross Ref Convergence analysis of Tikhonov regularization for non-linear statistical inverse problemsElectronic Journal of Statistics, Vol. 14, No. 2 Cross Ref Concentration of weakly dependent Banach-valued sums and applications to statistical learning methodsBernoulli, Vol. 25, No. 4B Cross Ref Optimal rates for coefficient-based regularized regressionApplied and Computational Harmonic Analysis, Vol. 47, No. 3 Cross Ref A mean-field optimal control formulation of deep learning13 December 2018 | Research in the Mathematical Sciences, Vol. 6, No. 1 Cross Ref Concentration inequalities for separately convex functionsBernoulli, Vol. 24, No. 4A Cross Ref Optimal Rates for Regularization of Statistical Inverse Learning Problems20 June 2017 | Foundations of Computational Mathematics, Vol. 18, No. 4 Cross Ref Optimal Rates for the Regularized Learning Algorithms under General Source Condition27 March 2017 | Frontiers in Applied Mathematics and Statistics, Vol. 3 Cross Ref A stochastic procedure to solve linear ill-posed problems8 March 2016 | Communications in Statistics - Theory and Methods, Vol. 46, No. 3 Cross Ref Optimal-order bounds on the rate of convergence to normality in the multivariate delta methodElectronic Journal of Statistics, Vol. 10, No. 1 Cross Ref On the Bennett–Hoeffding inequalityAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques, Vol. 50, No. 1 Cross Ref Multiparameter Regularization for Construction of Extrapolating Estimators in Statistical Learning Theory12 August 2012 Cross Ref Optimal Re-centering Bounds, with Applications to Rosenthal-type Concentration of Measure Inequalities1 April 2013 Cross Ref Integral Operator Approach to Learning Theory with Unbounded Sampling17 February 2011 | Complex Analysis and Operator Theory, Vol. 6, No. 3 Cross Ref LEAST SQUARE REGRESSION WITH COEFFICIENT REGULARIZATION BY GRADIENT DESCENT1 May 2012 | International Journal of Wavelets, Multiresolution and Information Processing, Vol. 10, No. 01 Cross Ref CROSS-VALIDATION BASED ADAPTATION FOR REGULARIZATION OPERATORS IN LEARNING THEORY20 November 2011 | Analysis and Applications, Vol. 08, No. 02 Cross Ref Elastic-net regularization in learning theoryJournal of Complexity, Vol. 25, No. 2 Cross Ref Kernel regression estimation in a Banach spaceJournal of Statistical Planning and Inference, Vol. 139, No. 4 Cross Ref On regularization algorithms in learning theoryJournal of Complexity, Vol. 23, No. 1 Cross Ref On normal domination of (super)martingalesElectronic Journal of Probability, Vol. 11, No. none Cross Ref Erratum and complements to "Berry–Esseen inequality for linear processes in Hilbert Spaces"Statistics & Probability Letters, Vol. 70, No. 2 Cross Ref Exact Asymptotics for Large Deviation Probabilities, with Applications Cross Ref Approximation et inégalités exponentielles pour les sommes de vecteurs aléatoires dépendantsComptes Rendus Mathematique, Vol. 334, No. 2 Cross Ref An Asymptotic Representation for the Likelihood Ratio for Multidimensional Samples with Discontinuous DesitiesI. S. Borisov and D. V. Mironov25 July 2006 | Theory of Probability & Its Applications, Vol. 45, No. 2AbstractPDF (172 KB)Représentation autorégressive de l'opérateur de covariance empirique d'un ARH(1). ApplicationsComptes Rendus de l'Académie des Sciences - Series I - Mathematics, Vol. 329, No. 6 Cross Ref On a majorization inequality for sums of independent random vectorsStatistics & Probability Letters, Vol. 19, No. 2 Cross Ref Inequalities for Distributions of Sums of Independent Random Vectors and Their Application to Estimating a DensityI. F. Pinelis17 July 2006 | Theory of Probability & Its Applications, Vol. 35, No. 3AbstractPDF (286 KB)Exact Exponential Bounds for Sums of Independent Random VariablesI. S. Pinelis and S. A. Utev17 July 2006 | Theory of Probability & Its Applications, Vol. 34, No. 2AbstractPDF (616 KB) Volume 30, Issue 1| 1986Theory of Probability & Its Applications History Submitted:29 April 1983Published online:28 July 2006 InformationCopyright © 1986 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1130013Article page range:pp. 143-148ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics