Improved nonuniform Berry--Esseen-type bounds

New nonuniform Berry-Esseen-type bounds for sums of inde- pendent random variables are obtained, motivated by recent studies con- cerning such bounds for nonlinear statistics. The proofs are based on the Chen-Shao concentration techniques within the framework of the Stein method.

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

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

[3]  A. Bikelis.  On estimates of the remainder term in the central limit theorem , 1966 .

[4]  V. V. Petrov,et al.  On an Estimate of the Remainder Term in the Central Limit Theorem , 1967 .

[5]  A. Nagaev.  Integral Limit Theorems Taking Large Deviations Into Account When Cramér’s Condition Does Not Hold. II , 1969 .

[6]  V. V. Petrov.  Sums of Independent Random Variables , 1975 .

[7]  V. V. Petrov.  Asymptotic Expansions in the Central Limit Theorem , 1975 .

[8]  L. Rozovskii.  Asymptotic Expansions in the Central Limit Theorem , 1976 .

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

[10]  Sergey Utev,et al.  Exact Exponential Bounds for Sums of Independent Random Variables , 1990 .

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

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

[13]  Qi-Man Shao,et al.  A non-uniform Berry–Esseen bound via Stein's method , 2001 .

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

[15]  Louis H. Y. Chen,et al.  Stein's method for normal approximation , 2005 .

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

[17]  I. Pinelis.  On the Bennett–Hoeffding inequality , 2009, 0902.4058.

[18]  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 .

[19]  I. Pinelis,et al.  Optimal-order bounds on the rate of convergence to normality in the multivariate delta method , 2009, 0906.0177.

[20]  I. Pinelis.  Exact lower bounds on the exponential moments of Winsorized and truncated random variables , 2010, 1001.2901.