Nonuniform estimates of convergence rate in the central limit theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 57 (2012) no. 1, pp. 62-97 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

@article{TVP_2012_57_1_a2,
     author = {Yu. S. Nefedova and I. G. Shevtsova},
     title = {Nonuniform estimates of convergence rate in the central limit theorem},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {62--97},
     year = {2012},
     volume = {57},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2012_57_1_a2/}
}
TY  - JOUR
AU  - Yu. S. Nefedova
AU  - I. G. Shevtsova
TI  - Nonuniform estimates of convergence rate in the central limit theorem
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2012
SP  - 62
EP  - 97
VL  - 57
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TVP_2012_57_1_a2/
LA  - ru
ID  - TVP_2012_57_1_a2
ER  - 
%0 Journal Article
%A Yu. S. Nefedova
%A I. G. Shevtsova
%T Nonuniform estimates of convergence rate in the central limit theorem
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2012
%P 62-97
%V 57
%N 1
%U http://geodesic.mathdoc.fr/item/TVP_2012_57_1_a2/
%G ru
%F TVP_2012_57_1_a2
Yu. S. Nefedova; I. G. Shevtsova. Nonuniform estimates of convergence rate in the central limit theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 57 (2012) no. 1, pp. 62-97. http://geodesic.mathdoc.fr/item/TVP_2012_57_1_a2/

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

[2] Bikyalis A., “O tochnosti approksimatsii raspredelenii summ nezavisimykh odinakovo raspredelennykh sluchainykh velichin normalnym raspredeleniem”, Litov. matem. sb., 11:2 (1971), 237–240 | Zbl

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

[4] Gavrilenko S. V., “Utochnenie neravnomernykh otsenok skorosti skhodimosti raspredelenii puassonovskikh sluchainykh summ k normalnomu zakonu”, Informatika i ee primeneniya, 5:1 (2011), 12–24 | MR

[5] Grigoreva M. E., Shevtsova I. G., “Utochnenie neravenstva Katsa–Berri–Esseena”, Informatika i ee primeneniya, 4:2 (2010), 75–82 | MR

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

[7] 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

[8] Korolev V. Yu., Bening V. E., Shorgin S. Ya., Matematicheskie osnovy teorii riska, Fizmatlit, M., 2007

[9] Korolev V. Yu., Popov S. V., “Utochnenie otsenok skorosti skhodimosti v tsentralnoi predelnoi teoreme pri otsutstvii momentov poryadkov, bólshikh vtorogo”, Teoriya veroyatn. i ee primen., 56:4 (2011), 797–805 | MR

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

[11] 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 | MR

[12] 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

[13] Meshalkin L. D., Rogozin B. A., “Otsenka rasstoyaniya mezhdu funktsiyami raspredeleniya po blizosti ikh kharakteristicheskikh funktsii i ee primenenie k tsentralnoi predelnoi teoreme”, Predelnye teoremy teorii veroyatnostei, izd-vo AN UzSSR, Tashkent, 1963, 49–55

[14] Mirakhmedov Sh. A., “Ob absolyutnoi postoyannoi v neravnomernoi otsenke skorosti skhodimosti v tsentralnoi predelnoi teoreme”, Izv. AN UzSSR, ser. fiz.-mat. nauk, 1984, no. 4, 26–30 | MR | Zbl

[15] Nagaev S. V., “Nekotorye predelnye teoremy dlya bolshikh uklonenii”, Teoriya veroyatnostei i ee primeneniya, 10:2 (1965), 231–254 | MR | Zbl

[16] Nefedova Yu. S., Shevtsova I. G., “O tochnosti normalnoi approksimatsii dlya raspredelenii puassonovskikh sluchainykh summ”, Informatika i ee primeneniya, 5:1 (2011), 39–45

[17] Nikulin V. N., “Neravnomernye otsenki ostatochnogo chlena v tsentralnoi predelnoi teoreme”, Teoriya veroyatn. i ee primen., 34:4 (1992), 831–832

[18] Osipov L. V., “Utochnenie teoremy Lindeberga”, Teoriya veroyatn. i ee primen., 11:2 (1966), 339–342 | 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] Paditts L., Mirakhmedov Sh. A., “Pismo v redaktsiyu (Zamechanie k otsenke absolyutnoi postoyannoi v neravnomernoi otsenke skorosti skhodimosti v ts.p.t.)”, Izv. AN UzSSR, ser. fiz.-mat. nauk, 1986, no. 3, 80 | MR

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

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

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

[24] Petrov V. V., “Ob otsenke ostatochnogo chlena v tsentralnoi predelnoi teoreme”, Zapiski nauchnykh seminarov POMI, 341 (2007), 142–146 | MR | Zbl

[25] Rozovskii L. V., “Neravnomernaya otsenka ostatochnogo chlena v tsentralnoi predelnoi teoreme”, Zapiski nauchnykh seminarov POMI, 351 (2007), 238–241 | MR

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

[27] Chistyakov G. P., “Ob odnoi zadache A. N. Kolmogorova”, Zapiski nauchnykh seminarov LOMI, 184 (1990), 289–319 | MR

[28] Shevtsova I. G., “O tochnosti normalnoi approksimatsii dlya raspredelenii puassonovskikh sluchainykh summ”, Obozrenie prikl. i promyshl. matem., 14:1 (2007), 3–28 | Zbl

[29] Shevtsova I. G., “Ob asimptoticheski pravilnykh postoyannykh v neravenstve Berri–Esseena–Katsa”, Teoriya veroyatnostei i ee primeneniya, 55:2 (2010), 271–304 | MR

[30] Bening V., Korolev V., Generalized Poisson Models and their Applications in Insurance and Finance, VSP, Utrecht, 2002 | Zbl

[31] 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

[32] Chen L. H. Y., Shao Q. M., “A non-uniform Berry–Esseen bound via Stein's method”, Probability Theory and Related Fields, 120 (2001), 236–254 | DOI | MR | Zbl

[33] Cramér H., “Sur un nouveau théorèm-limite de la théorie des probabilités”, Actualités scientifiques et industrielles, 736, Paris, 1938 | Zbl

[34] 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

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

[36] 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

[37] Hoeffding W., “The extrema of the expected value of a function of independent random variables”, Ann. Math. Statist., 19 (1948), 239–325 | MR

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

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

[40] Korolev V., Shevtsova I., “An improvement of the Berry–Esseen inequality with applications to Poisson and mixed Poisson random sums”, Scandinavian Actuarial Journal, 2010 ; Online first: 04 June 2010 http://www.informaworld.com/10.1080/03461238.2010.485370 | MR

[41] Korolev V. Yu., Popov S. V., “Utochnenie otsenok skorosti skhodimosti v tsentralnoi predelnoi teoreme pri oslablennykh momentnykh usloviyakh”, Dokl. RAN, 2012 (to appear)

[42] Michel R., “On the accuracy of nonuniform Gaussian approximation to the distribution functions of sums of independent and identically distributed random variables”, Z. Wahrsch. verw. Geb., 35:4 (1976), 337–347 | DOI | MR | Zbl

[43] Michel R., “On the constant in the nonuniform version of the Berry–Esseen theorem”, Z. Wahrsch. verw. Geb., 55 (1981), 109–117 | DOI | MR | Zbl

[44] Michel R., “On Berry–Esseen results for the compound Poisson distribution”, Insurance: Mathematics and Economics, 13:1 (1993), 35–37 | DOI | MR | Zbl

[45] Nakata T., “A nonuniform bound on convergence to normality for independent random variables”, Advances in Applied Probability, 11:2 (1977), 285–286 | DOI

[46] Neammanee K., “On the constant in the nonuniform version of the Berry–Esseen theorem”, International Journal of Mathematics and Mathematical Sciences, 12 (2005), 1951–1967 | DOI | MR | Zbl

[47] Neammanee K., Thongtha P., “Improvement of the non-uniform version of Berry-Esseen inequality via Paditz–Siganov theorems”, Journal of Inequalities in Pure and Applied Mathematics, 8:4 (2007), 92 | MR | Zbl

[48] Nikulin V., An algorithm to estimate a nonuniform convergence bound in the central limit theorem, 2010 http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.0552v1.pdf

[49] Nikulin V., Paditz L., “A note on nonuniform CLT–bounds”, 7th Vilnius Conference on Probability Theory and 22nd European Meeting of Statisticians, Abstracts, 1998, 358–359

[50] Paditz L., “Abschätzungen der Konvergenzgeschwindigkeit im zentralen Grenzwertsatz”, Wiss. Z. der TU Dresden, 25 (1976), 1169–1177 | MR | Zbl

[51] Paditz L., Über die Annäherung der Verteilungsfunktionen von Summen unabhängiger Zufallsgrößen gegen unbegrenzt teilbare Verteilungsfunktionen unter besonderer beachtung der Verteilungsfunktion der standardisierten Normalverteilung, Dissertation A, Technische Universität Dresden, Dresden, 1977

[52] Paditz L., “Abschätzungen der Konvergenzgeschwindigkeit zur Normalverteilung unter Voraussetzung einseitiger Momente”, Math. Nachr., 82 (1978), 131–156 | DOI | MR | Zbl

[53] Paditz L., “Über eine Fehlerabschätzung im zentralen Grenzwertsatz”, Wiss. Z. der TU Dresden, 28:5 (1979), 1197–1200 | MR | Zbl

[54] Paditz L., “Bemerkungen zu einer Fehlerabschätzung im zentralen Grenzwertsatz”, Wiss. Z. Hochschule für Verkehrswesen “Friedrich List”, 27:4 (1980), 829–837 | MR | Zbl

[55] Paditz L., “Einseitige Fehlerabschätzungen im zentralen Grenzwertsatz”, Math. Operationsforsch. und Statist., ser. Statist., 12 (1981), 587–604 | MR | Zbl

[56] Paditz L., “On error–estimates in the central limit theorem for generalized linear discounting”, Math. Operationsforsch. u. Statist., Ser. Statistics, 15:4 (1984), 601–610 | MR | Zbl

[57] Paditz L., “Über eine Fehlerabschätzung im zentralen Grenzwertsatz”, Wiss. Z. Hochschule für Verkehrswesen “Friedrich List”, 33, no. 2, Dresden, 1986, 399–404 | MR | Zbl

[58] Paditz L., “On the analytical structure of the constant in the nonuniform version of the Esseen inequality”, Statistics, 20:3 (1989), 453–464 | DOI | MR | Zbl

[59] Rychlik Z., “Nonuniform central limit bounds and their applications”, Teoriya veroyatn. i ee primen., 28:4 (1983), 646–652 | MR | Zbl

[60] Thongtha P., Neammanee K., “Refinement on the constants in the non-uniform version of the Berry–Esseen theorem”, Thai Journal of Mathematics, 5 (2007), 1–13 | MR | Zbl

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