Sobolev, V. N.; Kondratenko, A. E. On Senatov moments in asymptotic expansions in the central limit theorem. (English. Russian original) Zbl 1492.60060 Theory Probab. Appl. 67, No. 1, 154-157 (2022); translation from Teor. Veroyatn. Primen. 67, No. 1, 193-198 (2022). Summary: Representations are put forward for the moments and the truncated Senatov quasi-moments of normalized sums of random variables (r.v.’s) in terms of the Senatov moments of the original distribution. These representations make possible the direct transition from new asymptotic expansions in the central limit theorem to Gram-Charlier type expansions and are applied in the new proof of formulas for the convergence rate of these moments. Cited in 1 Document MSC: 60F05 Central limit and other weak theorems Keywords:central limit theorem; asymptotic expansions; Edgeworth-Cramér expansions; Gram-Charlier expansions; Senatov moments; Senatov quasi-moments × Cite Format Result Cite Review PDF Full Text: DOI References: [1] H. Cramér, On the composition of elementary errors: First paper: Mathematical deductions, Scand. Actuar. J., 1928 (1928), pp. 13-74, https://doi.org/10.1080/03461238.1928.10416862; Second paper: Statistical applications, 1928 (1928), pp. 141-180, https://doi.org/10.1080/03461238.1928.10416872. · JFM 54.0557.02 [2] C.-G. Esseen, Fourier analysis of distribution functions: A mathematical study of the Laplace-Gaussian law, Acta Math., 77 (1945), pp. 1-125, https://doi.org/10.1007/BF02392223. · Zbl 0060.28705 [3] B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, MA, 1954. · Zbl 0056.36001 [4] A. E. Kondratenko, The relation between a rate of convergence of moments of normed sums and the Chebyshev-Hermite moments, Theory Probab. Appl., 46 (2002), pp. 352-355, https://doi.org/10.1137/S0040585X97979007. · Zbl 1001.60029 [5] V. V. Senatov, Application of the Chebyshev-Hermite moments in asymptotic decompositions, in “Twentieth International Seminar on Stability Problems for Stochastic Models,” Theory Probab. Appl., 46 (2002), pp. 179-181, https://doi.org/10.1137/S0040585X97978853. [6] V. V. Senatov, The Central Limit Theorem: Accuracy of Approximation and Asymptotic Expansions, Knizhnyi Dom LIBROKOM, Moscow, 2009 (in Russian). [7] V. V. Senatov and V. N. Sobolev, New forms of asymptotic expansions in the central limit theorem, Theory Probab. Appl., 57 (2013), pp. 82-96, https://doi.org/10.1137/S0040585X97985807. · Zbl 1268.60034 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.