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Constants in Esseen–Rozovskii type inequalities proposed by Gabdullin, Makarenko, and Shevtsova (2018) are improved for one of the most important special cases: independent identically distributed random summands. The considered inequalities are used to estimate the uniform distance between the distribution function of the standardized sum of independent identically distributed random summands with finite second moments and the standard normal distribution function. As in the work by Gabdullin, Makarenko, and Shevtsova (2018), an estimate is made in terms of a truncated algebraic third-order moment and the tail of the second-order moment of the general distribution of random summands. [ABSTRACT FROM AUTHOR]

We prove two estimates of the rate of convergence in the Lindeberg theorem, involving algebraic truncated third-order moments and the classical Lindeberg fraction, which generalize a series of inequalities due to Esseen (Arkiv För Matematik8, 1, 7–15, 1969), Rozovskii (Bulletin of Leningrad University (in Russian), (1):70–75, 1974), & Wang and Ahmad (Sankhya A: Indian J.Stat.78, 2, 180–187, 2016), some of our recent results in Gabdullin, Makarenko, Shevtsova, (J. Math Sci.234, 6, 847–885, 2018, J. Math Sci.237, 5, 646–651, 2019b) and, up to constant factors, also Katz (Ann. Math. Statist.34, 1107–1108, 1963), Petrov (Soviet Math. Dokl.6, 5, 242–244, 1965), & Ibragimov and Osipov (Theory Probab. Appl.11, 1, 141–143, 1966b). The technique used in the proof is completely different from that in Wang and Ahmad (Sankhya A: Indian J.Stat.78, 2, 180–187, 2016) and is based on some extremal properties of introduced fractions which has not been noted in Katz (Ann. Math. Statist.34, 1107–1108, 1963), Petrov (Soviet Math. Dokl.6, 5, 242–244, 1965), & Wang and Ahmad (Sankhya A: Indian J.Stat.78, 2, 180–187, 2016). [ABSTRACT FROM AUTHOR]

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 improve bounds of accuracy of the normal approximation to the distribution of a sum of independent random variables under relaxed moment conditions, in particular, under the absence of moments of orders higher than the second. We extend these results to Poisson binomial, binomial, and Poisson random sums. Under the same conditions, we obtain bounds for the accuracy of approximation of the distributions of mixed Poisson random sums by the corresponding limit law. In particular, we construct these bounds for the accuracy of approximation of the distributions of geometric, negative binomial, and Poisson-inverse gamma (Sichel) random sums by the Laplace, variance gamma, and Student distributions, respectively. [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]

Upper estimates are presented for the universal constant in the Katz-Petrov and Osipov inequalities which do not exceed 3.1905. [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]

Many crucial results of the asymptotic theory of symmetric convex bodies were extended to the non-symmetric case in recent years. That led to the conjecture that for everyn-dimensional convex bodyK there exists a projectionP of rankk, proportional ton, such thatPK is almost symmetric. We prove that the conjecture does not hold. More precisely, we construct ann-dimensional convex bodyK such that for everyk >C√nlnn and every projectionP of rankk, the bodyPK is very far from being symmetric. In particular, our example shows that one cannot expect a formal argument extending the “symmetric” theory to the general case. [ABSTRACT FROM AUTHOR]

Following (Shevtsova, 2013) we introduce detailed classification of the asymptotically exact constants in natural estimates of the rate of convergence in the Lindeberg central limit theorem, namely in Esseen's, Rozovskii's, and Wang–Ahmad's inequalities and their structural improvements obtained in our previous works. The above inequalities involve algebraic truncated third-order moments and the classical Lindeberg fraction and assume finiteness only the second-order moments of random summands. We present lower bounds for the introduced asymptotically exact constants as well as for the universal and for the most optimistic constants which turn to be not far from the upper ones. [ABSTRACT FROM AUTHOR]