Theory of Probability and Mathematical Statistics
A sharper Lyapunov–Katz central limit error bound for i.i.d. summands Zolotarev-close to normal
Lena Jonas and Lutz Mattner
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Abstract: We prove a central limit error bound for convolution powers of laws with finite moments of order r∈]2,3], taking a closeness of the laws to normality into account. Up to a universal constant, this generalises the case of r=3 of the sharpening of the Berry (1941) – Esseen (1942) theorem obtained by Mattner (2024), namely by sharpening here the Katz (1963) error bound for the i.i.d. case of Lyapunov’s (1901) theorem.
Our proof uses a partial generalisation of the theorem of Senatov and Zolotarev (1986) used for the earlier special case. A result more general than our main one could be obtained by using instead another theorem of Senatov (1980), but, unfortunately, an auxiliary inequality used in the latter’s proof is wrong.
Keywords: Central limit theorem, sums of independent random variables
Bibliography: • Carl-Gustav Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Math. 77 (1945), 1–125. MR 14626, DOI 10.1007/BF02392223
• C. G. Esseen, A moment inequality with an application to the central limit theorem, Skand. Aktuarietidskr. 39 (1956), 160–170 (1957). MR 90166, DOI 10.1080/03461238.1956.10414946
• Larry Goldstein, Bounds on the constant in the mean central limit theorem, Ann. Probab. 38 (2010), no. 4, 1672–1689. MR 2663641, DOI 10.1214/10-AOP527
• L. Jonas, An estimate for the rate of convergence in the central limit theorem using Zolotarev’s distances for probability measures, unpublished master’s thesis, Universität Trier, 2024.
• Melvin L. Katz, Note on the Berry-Esseen theorem, Ann. Math. Statist. 34 (1963), 1107–1108. MR 151996, DOI 10.1214/aoms/1177704037
• Victor Korolev and Alexandra Dorofeeva, Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions, Lith. Math. J. 57 (2017), no. 1, 38–58. MR 3621871, DOI 10.1007/s10986-017-9342-7
• J. W. Lindeberg, Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung, Math. Z. 15 (1922), no. 1, 211–225 (German). MR 1544569, DOI 10.1007/BF01494395
• A. Lyapunov, Nouvelle forme du théorème sur la limite de probabilité, Zapiski Imperatorskoĭ akademīi nauk, po Fiziko-matematicheskomu otdielenīiu. 8e série = Mémoires de l’Académie impériale des sciences de St.-Pétersbourg, Classe physico-mathématique. 8e série 12 (1901), no. 5, 1–24 (French), https://www.biodiversitylibrary.org/page/57713987\punctskip.
• V. K. Matskyavichyus, A lower bound for the rate of convergence in the central limit theorem, Teor. Veroyatnost. i Primenen. 28 (1983), no. 3, 565–569 (Russian, with English summary). MR 716313
• Lutz Mattner, A convolution inequality, yielding a sharper Berry-Esseen theorem for summands Zolotarev-close to normal, Theory Probab. Math. Statist. 111 (2024), 45–122. MR 4823518, DOI 10.1090/tpms/1217
• Lutz Mattner and Irina Shevtsova, An optimal Berry-Esseen type theorem for integrals of smooth functions, ALEA Lat. Am. J. Probab. Math. Stat. 16 (2019), no. 1, 487–530. MR 3941867, DOI 10.30757/alea.v16-19
• L. V. Osipov and V. V. Petrov, On the estimation of the remainder term in the central limit theorem, Teor. Verojatnost. i Primenen. 12 (1967), 322–329 (Russian, with English summary). MR 216552
• Valentin V. Petrov, Limit theorems of probability theory, Oxford Studies in Probability, vol. 4, The Clarendon Press, Oxford University Press, New York, 1995. Sequences of independent random variables; Oxford Science Publications. MR 1353441
• V. V. Senatov, Some uniform estimates of the convergence rate in the multidimensional central limit theorem, Teor. Veroyatnost. i Primenen. 25 (1980), no. 4, 757–770 (Russian, with English summary). MR 595137
• Vladimir V. Senatov, Normal approximation: new results, methods and problems, Modern Probability and Statistics, VSP, Utrecht, 1998. Translated from the Russian manuscript by A. V. Kolchin. MR 1686374, DOI 10.1515/9783110933666.363
• I. G. Shevtsova, On asymptotically exact constants in the Berry-Esseen-Kac inequality, Teor. Veroyatn. Primen. 55 (2010), no. 2, 271–304 (Russian, with Russian summary); English transl., Theory Probab. Appl. 55 (2011), no. 2, 225–252. MR 2768905, DOI 10.1137/S0040585X97984772
• Irina Shevtsova, On the absolute constants in Nagaev-Bikelis-type inequalities, Inequalities and extremal problems in probability and statistics, Academic Press, London, 2017, pp. 47–102. MR 3702301
• I. S. Tyurin, On the rate of convergence in Lyapunov’s theorem, Teor. Veroyatn. Primen. 55 (2010), no. 2, 250–270 (Russian, with Russian summary); English transl., Theory Probab. Appl. 55 (2011), no. 2, 253–270. MR 2768904, DOI 10.1137/S0040585X97984760
• I. S. Tyurin, Some optimal bounds in the central limit theorem using zero biasing, Statist. Probab. Lett. 82 (2012), no. 3, 514–518. MR 2887466, DOI 10.1016/j.spl.2011.11.010
• V. M. Zolotarev, Letter to the editors: “Approximation of the distributions of sums of independent random variables with values in infinite-dimensional spaces” (Teor. Verojatnost. i Primenen. 21 (1976), no. 4, 741–758), Teor. Verojatnost. i Primenen. 22 (1977), no. 4, 901 (Russian). MR 517339
• V. M. Zolotarev, Metric distances in spaces of random variables and of their distributions, Mat. Sb. (N.S.) 101(143) (1976), no. 3, 416–454, 456 (Russian). MR 467869
• V. M. Zolotarev, Sovremennaya teoriya summirovaniya nezavisimykh sluchaĭ nykh velichin, Teoriya Veroyatnosteĭ i Matematicheskaya Statistika. [Probability Theory and Mathematical Statistics], “Nauka”, Moscow, 1986 (Russian). MR 917274
• Vladimir M. Zolotarev, Modern theory of summation of random variables, Modern Probability and Statistics, VSP, Utrecht, 1997. MR 1640024, DOI 10.1515/9783110936537
