Abstract
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 Matematik 8, 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).
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References
Agnew, R. P. (1957). Estimates for global central limit theorems. Ann. Math. Stat. 28, 26–42.
Berry, A. C. (1941). The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 49, 122–136.
Bhattacharya, R. N. and Ranga Rao, R. (1976). Normal approximation and asymptotic expansions. Wiley, New York.
Chebotarev, V. I., Kondrik, A. S. and Mikhailov, K. V. (2007). On an extreme two-point distribution. arXiv:0710.3456.
Esseen, C. -G. (1942). On the Liapounoff limit of error in the theory of probability. Ark. Mat. Astron. Fys. A28, 9, 1–19.
Esseen, C. -G. (1969). On the remainder term in the central limit theorem. Arkiv För Matematik 8, 1, 7–15.
Feller, W. (1935). ÜBer den zentralen Genzwertsatz der Wahrscheinlichkeitsrechnung. Math. Z. 40, 521–559.
Gabdullin, R., Makarenko, V. and Shevtsova, I. (2018). Esseen–Rozovskii type estimates for the rate of convergence in the Lindeberg theorem. J. Math Sci.234, 6, 847–885.
Gabdullin, R., Makarenko, V. and Shevtsova, I. (2019a). A generalization of the Rozovskii inequality. J. Math Sci. 237, 6, 775–781.
Gabdullin, R., Makarenko, V. and Shevtsova, I. (2019b). A generalization of the Wang–Ahmad inequality. J. Math Sci. 237, 5, 646–651.
Gabdullin, R. A., Makarenko, V. A. and Shevtsova, I. G. (2020). Asymptotically exact constants in natural convergence rate estimates in the Lindeberg theorem, to appear.
Ibragimov, I. A. (1966a). On the accuracy of the approximation of distribution functions of sums of independent variables by the normal distribution. Theory Probab Appl. 11, 4, 559–579.
Ibragimov, I. A. and Osipov, L. V. (1966b). On an estimate of the remainder in Lindeberg’s theorem. Theory Probab. Appl. 11, 1, 141–143.
Katz, M. L. (1963). Note on the Berry–Esseen theorem. Ann. Math. Statist. 34, 1107–1108.
Kondrik, A. S., Mikhailov, K. V. and Chebotarev, V. I. (2006). On the uniform estimate of the difference of distribution functions. In Abstracts of Communications of XXXI Far-East Academician E. V. Zolotov School-Seminar, pages 16–17, Vladivostok, Russia.
Korolev, V. Y. u. and Popov, S. V. (2012). Improvement of convergence rate estimates in the central limit theorem under weakened moment conditions. Dokl Math.86, 1, 506–511.
Korolev, V. and Dorofeyeva, A. (2017). Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions. Lith. Math. J. 57, 1, 38–58.
Lindeberg, J. W. (1922). Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung. Math. Z. 15, 1, 211–225.
Loh, W. Y. (1975). On the normal approximation for sums of mixing random variables. Master thesis, Department of Mathematics University of Singapore.
Osipov, L. V. (1966). Refinement of Lindeberg’s theorem. Theory Probab. Appl. 11, 2, 299–302.
Paditz, L. (1984). On error–estimates in the central limit theorem for generalized linear discounting. Math. Operationsforsch. u. Statist., Ser. Statistics 15, 4, 601–610.
Petrov, V. V. (1965). An estimate of the deviation of the distribution function of a sum of independent random variables from the normal law. Soviet Math. Dokl. 6, 5, 242–244.
Rozovskii, L. V. (1974). On the rate of convergence in the Lindeberg–Feller theorem. Bulletin of Leningrad University (in Russian), (1):70–75.
Shevtsova, I. G. (2013). On the absolute constant in the Berry–Esseen inequality and its structural and non-uniform improvements. Informatics and its Applications (in Russian), 7, 1, 124–125.
Wang, N. and Ahmad, I. A. (2016). A Berry–Esseen inequality without higher order moments. Sankhya A: Indian J.Stat. 78, 2, 180–187.
Zolotarev, V. M. (1997). Modern theory of summation of random variables. VSP, Utrecht, The Netherlands.
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Research on Theorem 1 was supported by the Russian Science Foundation (Project No. 18-11-00155). The rest part of the paper was supported by the Russian Foundation for Basic Research (project 19-07-01220-a) and by the Ministry for Education and Science of Russia (grant No. MD–189.2019.1).
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Gabdullin, R., Makarenko, V. & Shevtsova, I. On Natural Convergence Rate Estimates in the Lindeberg Theorem. Sankhya A 84, 671–688 (2022). https://doi.org/10.1007/s13171-020-00206-3
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DOI: https://doi.org/10.1007/s13171-020-00206-3
Keywords
- Central limit theorem
- Normal approximation
- Lindeberg’s condition
- Natural convergence rate estimate
- Truncated moment
- Absolute constant