Adopting ideas of Katz (1963), Petrov (1965), Wang and Ahmad (2016), and Gabdullin, Makarenko, and Shevtsova (2016), we generalize the Rozovskii inequality (1974) which provides an estimate of the accuracy of the normal approximation to distribution of a sum of independent random variables in terms of the absolute value of the sum of truncated in a fixed point third-order moments and the sum of the second-order tails of random summands. The generalization is due to introduction of a truncation parameter and a weighting function from a set of functions originally introduced by Katz (1963). The obtained inequality does not assume finiteness of moments of random summands of order higher than the second and may be even sharper than the celebrated inequalities of Berry (1941), Esseen (1942, 1969), Katz (1963), Petrov (1965), and Wang & Ahmad (2016).
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References
A. C. Berry, “The accuracy of the Gaussian approximation to the sum of independent variates,” Trans. Am. Math. Soc., 49, 122–136 (1941).
R. A. Gabdullin, V. A. Makarenko, and I. G. Shevtsova, “Esseen–Rozovskii type estimates for the rate of convergence in the Lindeberg theorem,” J. Math. Sci., 234, No. 6, 847–885 (2018).
R. A. Gabdullin, V. A. Makarenko, and I. G. Shevtsova, “A generalization of the Wang–Ahmad inequality,” J. Math. Sci., 237, No. 5, 646–651 (2019).
C.-G. Esseen, “On the Liapounoff limit of error in the theory of probability,” Ark. Mat. Astron. Fys., A28, No. 9, 1–19 (1942).
C.-G. Esseen, “Fourier analysis of distribution functions. A mathematical study of the Laplace–Gaussian law,” Acta Math., 77, No. 1, 1–125 (1945).
C.-G. Esseen, “On the remainder term in the central limit theorem,” Ark. Mat., 8, No. 1, 7–15 (1969).
W. Feller, “¨ Uber den zentralen Genzwertsatz der Wahrscheinlichkeitsrechnung,” Math. Z., 40, 521–559 (1935).
M. L. Katz, “Note on the Berry–Esseen theorem,” Ann. Math. Stat., 34, 1107–1108 (1963).
V. Korolev and A. Dorofeyeva, “Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions,” Lith. Math. J., 57, No. 1, 38–58 (2017).
V. Yu. Korolev and S. V. Popov, “Improvement of convergence rate estimates in the central limit theorem under weakened moment conditions,” Dokl. Math., 86, No. 1, 506–511 (2012).
J. W. Lindeberg, “Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung,” Math. Z., 15, No. 1, 211–225 (1922).
W. Y. Loh, On the normal approximation for sums of mixing random variables, Master Thesis, Department of Mathematics, University of Singapore (1975).
L. V. Osipov, “Refinement of Lindeberg’s theorem,” Theor. Probab. Appl., 10, No. 2, 299–302 (1966).
L. Paditz, “On error-estimates in the central limit theorem for generalized linear discounting,” Math. Operationsforsch. Stat., Ser. Stat., 15, No. 4, 601–610 (1984).
V. V. Petrov, “An estimate of the deviation of the distribution function of a sum of independent random variables from the normal law,” Sov. Math. Dokl., 6, No. 5, 242–244 (1965).
L. V. Rozovskii, “On the rate of convergence in the Lindeberg–Feller theorem,” Bull. Leningr. Univ., No. 1, 70–75 (1974).
I. G. Shevtsova, “On the absolute constant in the Berry–Esseen inequality and its structural and non-uniform improvements,” Informatics Appl., 7, No. 1, 124–125 (2013).
N. Wang and I. A. Ahmad, “A Berry–Esseen inequality without higher order moments,” Indian J. Stat., 78, No. 2, 180–187 (2016).
V. M. Zolotarev, Modern Theory of Summation of Random Variables, VSP, Utrecht, The Netherlands (1997).
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Proceedings of the XXXIV International Seminar on Stability Problems for Stochastic Models, Debrecen, Hungary.
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Gabdullin, R.A., Makarenko, V. & Shevtsova, I.G. A Generalization of the Rozovskii Inequality. J Math Sci 237, 775–781 (2019). https://doi.org/10.1007/s10958-019-04203-2
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DOI: https://doi.org/10.1007/s10958-019-04203-2