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Estimates are constructed for the deviation of the concentration functions of sums of independent random variables with finite variances from the folded normal distribution function without any assumptions concerning the existence of the moments of summands of higher orders. The results obtained are extended to Poisson-binomial, binomial, and Poisson random sums. Under the same assumptions, bounds are obtained for the approximation of the concentration functions of mixed Poisson random sums by the corresponding limit distributions. In particular, bounds are put forward for the accuracy of approximation of the concentration functions of geometric, negative binomial, and Sichel random sums by exponential, folded variance gamma, and folded Student distributions. Numerical estimates of all the constants involved are written down explicitly. [ABSTRACT FROM AUTHOR]

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]

Upper bounds of the absolute constant in the convergence rate estimate in the central limit theorem are sharpened in terms of truncated moments. It is shown that the absolute constant in the Osipov inequality does not exceed 2.011. [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]

A new proof of the absolute convergence of the Spitzer series is given which is based on the Berry-Esseen bound. Moreover, the upper bound is deduced for the sum of the series generated by the absolute values of the terms of the Spitzer series. [ABSTRACT FROM AUTHOR]