On the Bennett–Hoeffding inequality

The well-known Bennett-Hoeffding bound for sums of independent random variables is refined, by taking into account truncated third moments, and at that also improved by using, instead of the class of all increasing exponential functions, the much larger class of all generalized moment functions f such that f and f" are increasing and convex. It is shown that the resulting bounds have certain optimality properties. Comparisons with related known bounds are given. The results can be extended in a standard manner to (the maximal functions of) (super)martingales.

[1]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[2]  G. Choquet.  Theory of capacities , 1954 .

[3]  G. Bennett.  Probability Inequalities for the Sum of Independent Random Variables , 1962 .

[4]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[5]  J. Krivine,et al.  Anneaux préordonnés , 1964 .

[6]  S. Nagaev.  Some Limit Theorems for Large Deviations , 1965 .

[7]  R. Phelps.  Lectures on Choquet's Theorem , 1966 .

[8]  Morris L. Eaton,et al.  A Note on Symmetric Bernoulli Random Variables , 1970 .

[9]  A. Prékopa.  On logarithmic concave measures and functions , 1973 .

[10]  Morris L. Eaton,et al.  A Probability Inequality for Linear Combinations of Bounded Random Variables , 1974 .

[11]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[12]  D. Freedman.  On Tail Probabilities for Martingales , 1975 .

[13]  Iosif Pinelis,et al.  A Problem on Large Deviations in a Space of Trajectories , 1981 .

[14]  G. Cassier,et al.  Problème des moments sur un compact de Rn et décomposition de polynômes a plusieurs variables , 1984 .

[15]  David Handelman,et al.  Positive polynomials and product type actions of compact groups , 1985 .

[16]  I. Pinelis,et al.  Estimates of the Moments of Sums of Independent Random Variables , 1985 .

[17]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[18]  A. Borovkov,et al.  Second-Order Approximation of Random Polygonal Lines in the Donsker-Prokhorov Invariance Principle , 1987 .

[19]  D. Handelman.  Representing polynomials by positive linear functions on compact convex polyhedra. , 1988 .

[20]  Majorization, exponential inequalities and almost sure behavior of vector-valued random variables , 1991 .

[21]  M. Hallin,et al.  Improved Eaton bounds for linear combinations of bounded random variables , 1993 .

[22]  I. Pinelis.  AN APPROACH TO INEQUALITIES FOR THE DISTRIBUTIONS OF INFINITE-DIMENSIONAL MARTINGALES , 1992 .

[23]  F. Götze,et al.  The Rate of Convergence for Multivariate Sampling Statistics , 1993 .

[24]  I. Pinelis.  Extremal Probabilistic Problems and Hotelling's $T^2$ Test Under a Symmetry Condition , 1994, math/0701806.

[25]  I. Pinelis.  OPTIMUM BOUNDS FOR THE DISTRIBUTIONS OF MARTINGALES IN BANACH SPACES , 1994, 1208.2200.

[26]  I. Pinelis.  On a majorization inequality for sums of independent random vectors , 1994 .

[27]  M. Talagrand.  The missing factor in Hoeffding's inequalities , 1995 .

[28]  George G. Roussas,et al.  Exponential probability inequalities with some applications , 1996 .

[29]  M. Talagrand.  New concentration inequalities in product spaces , 1996 .

[30]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[31]  Robert M. Corless,et al.  A sequence of series for the Lambert W function , 1997, ISSAC.

[32]  I. Pinelis.  Optimal Tail Comparison Based on Comparison of Moments , 1998 .

[33]  V. Peña.  A General Class of Exponential Inequalities for Martingales and Ratios , 1999 .

[34]  Hadas Shachnai,et al.  Optimal Bounds on Tail Probabilities: A Study of an Approach , 1999 .

[35]  S. Boucheron,et al.  A sharp concentration inequality with applications , 1999, Random Struct. Algorithms.

[36]  P. Massart,et al.  About the constants in Talagrand's concentration inequalities for empirical processes , 2000 .

[37]  S. Boucheron,et al.  A sharp concentration inequality with applications , 1999, Random Struct. Algorithms.

[38]  K. Dzhaparidze,et al.  On Bernstein-type inequalities for martingales , 2001 .

[39]  Sergey G. Bobkov,et al.  On Gaussian and Bernoulli covariance representations , 2001 .

[40]  Iosif Pinelis,et al.  L'Hospital type rules for monotonicity: Applications to probability inequalities for sums of bounded random variables. , 2002 .

[41]  O. Bousquet.  A Bennett concentration inequality and its application to suprema of empirical processes , 2002 .

[42]  I. Pinelis.  Spherically symmetric functions with a convex second derivative and applications to extremal probabilistic problems , 2002 .

[43]  I. Pinelis.  Dimensionality Reduction in Extremal Problems for Moments of Linear Combinations of Vectors with Random Coefficients , 2003 .

[44]  I. Borisov.  MOMENT INEQUALITIES CONNECTED WITH ACCOMPANYING POISSON LAWS IN ABELIAN GROUPS , 2003 .

[45]  O. Bousquet.  Concentration Inequalities for Sub-Additive Functions Using the Entropy Method , 2003 .

[46]  V. Bentkus.  An Inequality for Tail Probabilities of Martingales with Differences Bounded from One Side , 2003 .

[47]  Svante Janson,et al.  Large deviations for sums of partly dependent random variables , 2004, Random Struct. Algorithms.

[48]  E. Ostrovsky,et al.  A Remark on the Inequalities of Bernstein - Markov Type in Exponential Orlicz and Lorentz Spaces , 2004 .

[49]  V. Bentkus.  On Hoeffding’s inequalities , 2004, math/0410159.

[50]  E. Rio,et al.  Concentration around the mean for maxima of empirical processes , 2005, math/0506594.

[51]  I. Pinelis.  On normal domination of (super)martingales , 2005, math/0512382.

[52]  J. Wellner,et al.  EMPIRICAL PROCESSES WITH APPLICATIONS TO STATISTICS (Wiley Series in Probability and Mathematical Statistics) , 1987 .

[53]  Iosif Pinelis,et al.  Exact inequalities for sums of asymmetric random variables, with applications , 2006 .

[54]  Iosif Pinelis,et al.  Toward the best constant factor for the Rademacher-Gaussian tail comparison , 2006 .

[55]  I. Pinelis.  Binomial upper bounds on generalized moments and tail probabilities of (super)martingales with differences bounded from above , 2005, math/0512301.

[56]  Convex concentration inequalities and forward-backward stochastic calculus , 2006 .

[57]  Iosif Pinelis,et al.  On l'Hospital-type rules for monotonicity. , 2006 .

[58]  I. Pinelis.  On inequalities for sums of bounded random variables , 2006, math/0603030.

[59]  V. Bentkus,et al.  On domination of tail probabilities of (super)martingales: Explicit bounds , 2006 .

[60]  Qi-Man Shao,et al.  Normal approximation for nonlinear statistics using a concentration inequality approach , 2007, 0708.4272.

[61]  V. V. Petrov,et al.  On Large Deviations of Sums of Independent Random Variables , 2007 .

[62]  Optimal two-value zero-mean disintegration of zero-mean random variables , 2008, 0803.2068.

[63]  V. Bentkus,et al.  Bounds for tail probabilities of martingales using skewness and kurtosis , 2011, 1111.6358.

[64]  Positive-Part Moments via the Fourier–Laplace Transform , 2009, 0902.4214.

[65]  I. Pinelis,et al.  Berry-Esseen bounds for general nonlinear statistics, with applications to Pearson's and non-central Student's and Hotelling's , 2009 .

[66]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .