On the asymptotically exact constants in the Berry–Esseen–Katz inequality

I. G. Shevtsova

Teoriâ veroâtnostej i ee primeneniâ, Tome 55 (2010) no. 2, pp. 271-304

Bibliographie

[1] Bikyalis A., “Otsenki ostatochnogo chlena v tsentralnoi predelnoi teoreme”, Litov. matem. sb., 6:3 (1966), 323–346 | Zbl

[2] Bkhattachariya R. N., Ranga Rao R., Approksimatsiya normalnym raspredeleniem, Nauka, M., 1982, 286 pp.

[3] Gaponova M. O., Korchagin A. Yu., Shevtsova I. G., “Ob absolyutnykh konstantakh v ravnomernoi otsenke tochnosti normalnoi approksimatsii dlya raspredelenii, ne imeyuschikh tretego momenta”, Sbornik statei molodykh uchenykh fakulteta VMK MGU, Vyp. 6, Maks Press, M., 2009, 81–89

[4] Gaponova M. O., Shevtsova I. G., “Asimptoticheskie otsenki absolyutnoi postoyannoi v neravenstve Berri–Esseena dlya raspredelenii, ne imeyuschikh tretego momenta”, Informatika i ee primeneniya, 3:4 (2009), 41–56

[5] Grigoreva M. E., Shevtsova I. G., “Utochnenie neravenstva Katsa–Berri–Esseena”, Informatika i ee primeneniya, 4:2 (2010) (to appear) | MR

[6] Zolotarev V. M., “Absolyutnaya otsenka ostatochnogo chlena v tsentralnoi predelnoi teoreme”, Teoriya veroyatn. i ee primen., 11:1 (1966), 108–119 | MR | Zbl

[7] Zolotarev V. M., “Nekotorye neravenstva teorii veroyatnostei i ikh primenenie k utochneniyu teoremy A. M. Lyapunova”, Dokl. AN SSSR, 177:3 (1967), 501–504 | MR | Zbl

[8] Zolotarev V. M., Sovremennaya teoriya summirovaniya nezavisimykh sluchainykh velichin, Nauka, M., 1986, 415 pp. | MR

[9] Ibragimov I. A., “O tochnosti approksimatsii funktsii raspredeleniya summ nezavisimykh velichin normalnym raspredeleniem”, Teoriya veroyatn. i ee primen., 11:4 (1966), 632–655 | Zbl

[10] Ibragimov I. A., Linnik Yu. V., Nezavisimye i statsionarno svyazannye velichiny, Nauka, M., 1965, 524 pp.

[11] Kolmogorov A. N., “Nekotorye raboty poslednikh let v oblasti predelnykh teorem teorii veroyatnostei”, Vestnik Mosk. un-ta, ser. fiz.-matem. i estestv. nauk, 7:10 (1953), 29–38 | Zbl

[12] Kondrik A. S., Mikhailov K. V., Chebotarev V. I., “O ravnomernoi otsenke raznosti funktsii raspredeleniya”, Tez. dokl. XXXI Dalnevostochnoi shkoly-seminara im. akad. E. V. Zolotova, Vladivostok, 2006, 16–17

[13] Korolev V. Yu., Shevtsova I. G., “O verkhnei otsenke absolyutnoi postoyannoi v neravenstve Berri–Esseena”, Teoriya veroyatn. i ee primen., 54:4 (2009), 671–695

[14] Korolev V. Yu., Shevtsova I. G., “Utochnenie neravenstva Berri–Esseena”, Dokl. RAN, 430:6 (2010), 738–742 | MR | Zbl

[15] Korolev V. Yu., Shevtsova I. G., “Utochnenie neravenstva Berri–Esseena s prilozheniyami k puassonovskim i smeshannym puassonovskim sluchainym summam”, Obozrenie prikl. i promyshl. matem., 17:1 (2010), 25–56

[16] Matskyavichyus V. K., “O nizhnei otsenke skorosti skhodimosti v tsentralnoi predelnoi teoreme”, Teoriya veroyatn. i ee primen., 28:3 (1983), 565–569 | MR | Zbl

[17] Nagaev S. V., Chebotarev V. I., “Novyi podkhod k otsenke absolyutnoi konstanty v neravenstve Berri–Esseena”, Tez. dokl. XXXI Dalnevostochnoi shkoly-seminara im. akad. E. V. Zolotova, Vladivostok, 2006, 19

[18] Osipov L. V., “O tochnosti priblizheniya raspredeleniya summy nezavisimykh sluchainykh velichin k normalnomu raspredeleniyu”, Dokl. AN SSSR, 178:5 (1968), 1013–1016 | MR | Zbl

[19] Osipov L. V., Petrov V. V., “Ob otsenke ostatochnogo chlena v tsentralnoi predelnoi teoreme”, Teoriya veroyatn. i ee primen., 12:2 (1967), 322–329 | MR

[20] Petrov V. V., “Odna otsenka otkloneniya raspredeleniya summy nezavisimykh sluchainykh velichin ot normalnogo zakona”, Dokl. AN SSSR, 160:5 (1965), 1013–1015 | MR | Zbl

[21] Petrov V. V., Summy nezavisimykh sluchainykh velichin, Nauka, M., 1972, 416 pp. | MR

[22] Petrov V. V., Predelnye teoremy dlya summ nezavisimykh sluchainykh velichin, Nauka, M., 1987, 320 pp. | MR

[23] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady. Elementarnye funktsii, Nauka, M., 1981, 798 pp. | MR | Zbl

[24] Rogozin B. A., “Odno zamechanie k rabote Esseena “Momentnoe neravenstvo s primeneniem k tsentralnoi predelnoi teoreme””, Teoriya veroyatn. i ee primen., 5:1 (1960), 125–128 | MR

[25] Rozovskii L. V., “Ob otsenke snizu ostatochnogo chlena v tsentralnoi predelnoi teoreme”, Matem. zametki, 24:3 (1978), 403–410 | MR | Zbl

[26] Rozovskii L. V., “O tochnosti otsenki ostatochnogo chlena v tsentralnoi predelnoi teoreme”, Teoriya veroyatn. i ee primen., 23:4 (1978), 744–761 | MR | Zbl

[27] Senatov V. V., Tsentralnaya predelnaya teorema: tochnost approksimatsii i asimptoticheskie razlozheniya, URSS, Knizhnyi dom “Librokom”, M., 2009, 352 pp.

[28] Tyurin I. S., “Utochnenie verkhnikh otsenok konstant v teoreme Lyapunova”, Uspekhi matem. nauk, 65:3 (2010), 201–202 | MR

[29] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 2, Mir, M., 1984, 752 pp. | MR

[30] Chistyakov G. P., “Novoe asimptoticheskoe razlozhenie i asimptoticheski nailuchshie postoyannye v teoreme Lyapunova. I, II, III”, Teoriya veroyatn. i ee primen., 46:2 (2001), 326–344 ; 3, 573–579 ; 47:3 (2002), 475–497 | MR | Zbl | MR | MR | Zbl

[31] Shevtsova I. G., “Utochnenie verkhnei otsenki absolyutnoi postoyannoi v neravenstve Berri–Esseena”, Teoriya veroyatn. i ee primen., 51:3 (2006), 622–626 | MR

[32] Shevtsova I. G., “Utochnenie absolyutnoi konstanty v klassicheskom neravenstve Berri–Esseena”, Statisticheskie metody otsenivaniya i proverki gipotez, 21, Izd-vo Permskogo gos. un-ta, Perm, 2008, 159–168

[33] Shevtsova I. G., “Nizhnyaya asimptoticheski pravilnaya postoyannaya v tsentralnoi predelnoi teoreme”, Dokl. RAN, 430:4 (2010), 466–469 | MR | Zbl

[34] Shiganov I. S., “Ob utochnenii verkhnei konstanty v ostatochnom chlene tsentralnoi predelnoi teoremy”, Problemy ustoichivosti stokhasticheskikh modelei, Trudy VNIISI, 1982, 109–115 | MR | Zbl

[35] van Beek P., “An application of Fourier methods to the problem of sharpening the Berry–Esseen inequality”, Z. Wahrscheinlichkeitstheor. verw. Geb., 23 (1972), 187–196 | DOI | MR | Zbl

[36] Bentkus V., On the asymptotical behaviour of the constant in the Berry–Esseen inequality, Preprint 91-078, Universität Bielefeld, Bielefeld, 1991

[37] Bentkus V., “On the asymptotical behavior of the constant in the Berry–Esseen inequality”, J. Theoret. Probab., 7:2 (1994), 211–224 | DOI | MR | Zbl

[38] Bergström H., “On the central limit theorem”, Skand. Aktuarietidskr., 27 (1944), 139–153 | MR | Zbl

[39] Bergström H., “On the central limit theorem in the case of not equally distributed random variables”, Skand. Aktuarietidskr., 32 (1949), 37–62 | MR

[40] Berry A. C., “The accuracy of the Gaussian approximation to the sum of independent variates”, Trans. Amer. Math. Soc., 49 (1941), 122–136 | MR

[41] Dunnage J. E. A., Review of: Prawitz H. On the remainder in the central limit theorem, 1695)}, Amer. Math. Soc., 1977, Mathematical Reviews on the Web MR 0397839(53{#

[42] Esseen C.-G., “On the Liapounoff limit of error in the theory of probability”, Ark. Mat. Astron. Fys., A28:9 (1942), 1–19 | MR | Zbl

[43] Esseen C.-G., “A moment inequality with an application to the central limit theorem”, Skand. Aktuarietidskr., 39 (1956), 160–170 | MR

[44] de Haan L., Peng L., “Slow convergence to normality: an Edgeworth expansion without third moment”, Probab. Math. Statist., 17:2 (1997), 395–406 | MR | Zbl

[45] Hall P., “On the rate of convergence in the central limit theorem for distributions with regularly varying tails”, Z. Wahrscheinlichkeitstheor. verw. Geb., 49:1 (1979), 1–11 | DOI | MR | Zbl

[46] Hall P., “Characterizing the rate of convergence in the central limit theorem”, Ann. Probab., 8:6 (1980), 1037–1048 | DOI | MR | Zbl

[47] Hall P., Rates of Convergence in the Central Limit Theorem., Res. Notes Math., 62, Pitman, Boston, London, Melbourne, 1982, 251 pp. | MR | Zbl

[48] Hall P., Barbour A. D., “Reversing the Berry–Esseen inequality”, Proc. Amer. Math. Soc., 90:1 (1984), 107–110 | MR | Zbl

[49] Heyde C. C., “On the influence of moments on the rate of convergence to the normal distribution”, Z. Wahrscheinlichkeitstheor. verw. Geb., 8:1 (1967), 12–18 | DOI | MR | Zbl

[50] Heyde C. C., “A nonuniform bound on convergence to normality”, Ann. Probab., 3:5 (1975), 903–907 | DOI | MR | Zbl

[51] Hoeffding W., “The extrema of the expected value of a function of independent random variables”, Ann. Math. Statist., 26:2 (1955), 268–275 | DOI | MR | Zbl

[52] Hsu P. L., “The approximate distributions of the mean and variance of a sample of independent variables”, Ann. Math. Statist., 16:1 (1945), 1–29 | DOI | MR | Zbl

[53] Ikeda S., “A note on the normal approximation to the sum of independent random variables”, Ann. Inst. Statist. Math., 11 (1959), 121–130 | DOI | MR | Zbl

[54] Katz M. L., “Note on the Berry–Esseen theorem”, Ann. Math. Statist., 34 (1963), 1107–1108 | DOI | MR | Zbl

[55] Korolev V., Shevtsova I., “An improvement of the Berry–Esseen inequality with applications to Poisson and mixed Poisson random sums”, Scand. Actuar. J., 2010 (to appear) | MR

[56] Paditz L., “Über eine Fehlerabschätzung im zentralen Grenzwertsatz”, Wiss. Z. Hochschule für Verkehrswesen “Friedrich List” Dresden, 33:2 (1986), 399–404 | MR | Zbl

[57] Prawitz H., “On the remainder in the central limit theorem. I: One-dimensional independent variables with finite absolute moments of third order”, Scand. Actuar. J., 3 (1975), 145–156 | MR

[58] Takano K. A., “Remark to a result of A. C. Berry”, Res. Mem. Inst. Math., 9:6 (1951), 4.08–4.15

[59] Tysiak W., Gleichmäßige und nicht-gleichmäßige Berry-Esseen-Abschätzungen, Dissertation, Gesamthochschule Wuppertal, Wuppertal, 1983 | Zbl

[60] Tyurin I., New estimates of the convergence rate in the Lyapunov theorem, 2009, arXiv: 0912.0726v1

[61] Uspensky J. V., Introduction to Mathematical Probability, McGraw-Hill, New York, 1937, 411 pp. | Zbl

[62] Wallace D. L., “Asymptotic approximations to distributions”, Ann. Math. Statist., 29 (1958), 635–654 | DOI | MR | Zbl

[63] Wallace D. L., A corrected computation of Berry's bound for the central limit theorem error, Statist. Res. Center, University of Chicago, Chicago, 1959

[64] Zahl S., “Bounds for the central limit theorem error”, SIAM J. Appl. Math., 14:6 (1966), 1225–1245 | DOI | MR | Zbl

[65] Zolotarev V. M., “A sharpening of the inequality of Berry–Esseen”, Z. Wahrscheinlichkeitstheor. verw. Geb., 8 (1967), 332–342 | DOI | MR | Zbl