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We sharpen the upper bounds for the absolute constant in nonuniform convergence rate estimates in the central limit theorem for sums of independent identically distributed random variables possessing absolute moments of the order 2 + δ with some 0 < δ 1. In particular, it is demonstrated that under the existence of the third moment this constant does not exceed 18.2. Also it is shown that the absolute constant in the estimates under consideration can be replaced by a function C∗(∣x∣, δ) of the argument x of the difference of the prelimit and limit normal distribution functions for which a positive bounded nonincreasing majorant is found. Moreover, for δ = 1 this majorant is asymptotically exact (unimprovable) as x→∞and sharpens the estimates due to Nikulin [preprint, arXiv:1004.0552v1 [math.ST], 2010] for all x. For the first time a similar result is obtained for the case δ ∈ (0, 1). As a corollary, we obtain upper estimates for the Kolmogorov functions which are the analogues of the exact and the asymptotically exact constants in the (uniform) Berry–Esseen inequality. [ABSTRACT FROM AUTHOR]

The inequality Δn≦0.3197·Σni=1(βi+σ³i) (Σni=1σ²i)-3/2 is proved for the uniform distance Δn between the distribution function of the standard normal law and the distribution function of the standardized sum of an arbitrary number n≧1 of independent random variables X1,…,Xn with zero means, variances σ²i= DXi>0, and finite third absolute moments βi = E ∣ Xi∣³, i =1,…n. [ABSTRACT FROM AUTHOR]