On the Nonuniform Berry-Esseen Bound

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© 2017 Elsevier Ltd. All rights reserved. Due to the effort of a number of authors, the value cu of the absolute constant factor in the uniform Berry-Esseen (BE) bound for sums of independent random variables has been gradually reduced to 0.4748 in the iid case and 0.5600 in the general case; both these values were recently obtained by Shevtsova. On the other hand, Esseen had shown that cu cannot be less than 0.4097. Thus, the gap factor between the best known upper and lower bounds on (the least possible value of) cu is now rather close to 1.The situation is quite different for the absolute constant factor cnu in the corresponding nonuniform BE bound. Namely, the best correctly established upper bound on cnu in the iid case is over 17 times the corresponding best known lower bound, and this gap factor is greater than 21 in the general case. In the present paper, improvements to the prevailing method (going back to S. Nagaev) of obtaining nonuniform BE bounds are suggested. Moreover, two new methods are presented, of a rather purely Fourier kind, based on two families of smoothing inequalities, which work better in the tail zones. As illustrations, two corresponding quick proofs of Nagaev's nonuniform BE bound are given. Some further refinements in the application of the methods are shown as well.

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Inequalities and Extremal Problems in Probability and Statistics: Selected Topics