Geodesic
On the accuracy of the normal approximation. I
Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 2, pp. 353-366 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Presented are practically applicable estimates of the accuracy of the normal approximation for the distributions of sums of independent identically distributed absolutely continuous random variables with finite moments of order $2+\delta$, $0<\delta\le 1$. The right-hand side of the estimate is the sum of two summands, the first being the Lyapunov fraction with the absolute constant arbitrarily close to the asymptotically exact one, whereas the second summand decreases exponentially fast.
Keywords: central limit theorem, normal approximation, Berry–Esseen inequality, convergence rate estimate, asymptotically exact constant. 
@article{TVP_2005_50_2_a8,
     author = {V. Yu. Korolev and I. G. Shevtsova},
     title = {On the accuracy of the normal {approximation.~I}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {353--366},
     year = {2005},
     volume = {50},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2005_50_2_a8/}
}
TY  - JOUR
AU  - V. Yu. Korolev
AU  - I. G. Shevtsova
TI  - On the accuracy of the normal approximation. I
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2005
SP  - 353
EP  - 366
VL  - 50
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TVP_2005_50_2_a8/
LA  - ru
ID  - TVP_2005_50_2_a8
ER  - 
%0 Journal Article
%A V. Yu. Korolev
%A I. G. Shevtsova
%T On the accuracy of the normal approximation. I
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2005
%P 353-366
%V 50
%N 2
%U http://geodesic.mathdoc.fr/item/TVP_2005_50_2_a8/
%G ru
%F TVP_2005_50_2_a8
V. Yu. Korolev; I. G. Shevtsova. On the accuracy of the normal approximation. I. Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 2, pp. 353-366. http://geodesic.mathdoc.fr/item/TVP_2005_50_2_a8/

[1] Bkhattachariya R. N., Ranga Rao R., Approksimatsiya normalnym raspredeleniem i asimptoticheskie razlozheniya, Nauka, M., 1982, 286 pp. | MR

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

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

[4] Loev M., Teoriya veroyatnostei, IL, M., 1962, 719 pp.

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

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

[7] Ppokhopov Yu. V., “Ob odnoi lokalnoi teopeme”, Ppedelnye teopemy teopii vepoyatnostei, ed. S. Kh. Sirazhdinov, Izd-vo AN UzSSR, Tashkent, 1963, 75–80

[8] 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

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

[10] Chistyakov G. P., “Novoe asimptoticheskoe razlozhenie i asimptoticheski nailuchshie postoyannye v teoreme Lyapunova. I”, Teoriya veroyatn. i ee primen., 46:2 (2001), 326–344 | MR | Zbl

[11] Shiganov I. S., “Ob utochnenii verkhnei konstanty v ostatochnom chlene tsentralnoi predelnoi teoremy”, Problemy ustoichivosti stokhasticheskikh modelei, eds. V. M. Zolotarev i V. V. Kalashnikov, VNIISI, M., 1982, 109–115 | MR

[12] 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

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

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

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

[16] Esseen C.-G., “Fourier analysis of distribution functions. A mathematical study of the Laplace–Gaussian law”, Acta Math., 77 (1945), 1–125 | DOI | MR | Zbl

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

[18] 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

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

[20] Paditz L., “On the error-bound in the nonuniform version of Esseen's inequality in the $L_p$-metric”, Statistics, 27:3–4 (1996), 379–394 | DOI | MR | Zbl

[21] Prawitz H., “Limits for a distribution, if the characteristic function is given in a finite domain”, Skand. Aktuarietidskr., 1972 (1973), 138–154 | MR | Zbl

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

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

[24] Ushakov N. G., Selected Topics in Characteristic Functions, VSP, Utrecht, 1999, 355 pp. | MR | Zbl

[25] 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

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

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