On the nonuniform Berry--Esseen bound

Original · EN

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 about 25 times the corresponding best known lower bound, and this gap factor is greater than 30 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, a new method is presented, of a rather purely Fourier kind, based on a family of smoothing inequalities, which work better in the tail zones. As an illustration, a quick proof of Nagaev's nonuniform BE bound is given. Some further refinements in the application of the method are shown as well.