Geodesic
Stability problems in Cramér-type characterization in case of I.I.D. Summands
Teoriâ veroâtnostej i ee primeneniâ, Tome 57 (2012) no. 4, pp. 701-723 Cet article a éte moissonné depuis la source Math-Net.Ru

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The stability property in Cramér’s characterization of the normal law is considered in the case of identically distributed summands. As opposite results, instability is shown with respect to strong distances including the entropic distance to normality (addressing a question of M. Kac).
Keywords: Cramér’s theorem; Cramér’s characterization of the normal law; stability problems. 
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S. G. Bobkov; G. P. Chistyakov; F. Götze. Stability problems in Cramér-type characterization in case of I.I.D. Summands. Teoriâ veroâtnostej i ee primeneniâ, Tome 57 (2012) no. 4, pp. 701-723. http://geodesic.mathdoc.fr/item/TVP_2012_57_4_a4/

[1] Bobkov S. G., Chistyakov G. P., Götze F., “Entropic instability of Cramér's characterization of the normal law”, Selected Works of Willem van Zwet, Sel. Works Probab. Statist., Springer, New York, 2012, 231–242 | MR

[2] Bobkov S. G., Chistyakov G. P., Götze F., Regularized distributions and entropic stability of Cramér's characterization of the normal law, Preprint, Bielefeld, 2010

[3] Chistyakov G. P., “O tochnosti otsenok v teoremakh ob ustoichivosti razlozhenii normalnogo raspredeleniya i raspredeleniya Puassona”, Teoriya funktsii, funktsionalnyi analiz i ikh prilozheniya, 1976, no. 26, 119–128, iii | Zbl

[4] Chistyakov G. P., “Zamechanie k teoreme N. A. Sapogova ob ustoichivosti razlozhenii normalnogo raspredeleniya”, Operatory v funktsionalnykh prostranstvakh i voprosy teorii funktsii, Naukova dumka, Kiev, 1987, 108–116 ; 147 | MR

[5] Chistyakov G. P., Golinskii L. V., “Otsenki ustoichivosti razlozhenii normalnogo raspredeleniya v metrike Levi”, Problemy ustoichivosti stokhasticheskikh modelei, VNIISI, M., 199, 16–40 | MR

[6] Christoph G., Prohorov Yu. V., Ulyanov V. V., “Characterization and stability problems for finite quadratic forms”, Asymptotic Methods in Probability and Statistics with Applications (St. Petersburg, 1998), Birkhäuser, Boston, 2001, 39–50 | DOI | MR | Zbl

[7] Cramér H., “Über eine Eigenschaft der Normalen Verteilungsfunktion”, Math. Z., 41 (1936), 405–414 | DOI | MR

[8] Dudley R. M., Real Analysis and Probability, Cambridge Univ. Press, Cambridge, 2002, 555 pp. | MR

[9] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 2, Mir, M., 1984, 751 pp.

[10] Ilin V. V., “O kolichestvennoi ustoichivosti teoremy Kramera v ravnomernoi metrike dlya odinakovo raspredelennykh sluchainykh velichin”, Dokl. AN SSSR, 260:3 (1981), 525–526 | MR

[11] Kagan A. M., Linnik Yu. V., Rao S. R., Kharakterizatsionnye zadachi matematicheskoi statistiki, Nauka, M., 1972, 656 pp. | MR

[12] Linnik Yu. V., Ostrovskii I. V., Razlozheniya sluchainykh velichin i vektorov, Nauka, M., 1972, 479 pp. | MR

[13] Maloshevskii S. G., “Neuluchshaemost rezultata N. A. Sapogova v probleme ustoichivosti teoremy G. Kramera”, Teoriya veroyatn. i ee primen., 13:3 (1968), 522–525

[14] Maloshevskii S. G., “Otsenki, svyazannye s problemoi ustoichivosti teoremy G. Kramera”, Matem. zametki, 7 (1970), 281–288

[15] McKean H. P., Jr., “Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas”, Arch. Ration. Mech. Anal., 21 (1966), 343–367 | DOI | MR

[16] Petrov V. V., Summy nezavisimykh sluchainykh velichin, Nauka, M., 1972, 414 pp. | MR

[17] Sapogov N. A., “Problema ustoichivosti dlya teoremy Kramera”, Izv. AN SSSR, 15:3 (1951), 205–218 | MR | Zbl

[18] Sapogov N. A., “Problema ustoichivosti dlya teoremy Kramera”, Vestn. Leningr. un-ta, 10:11 (1955), 61–64 | MR | Zbl

[19] Senatov V. V., “Ob utochnenii otsenok ustoichivosti dlya teoremy G. Kramera”, Zap. nauch. sem. LOMI, 61, 1976, 125–134 | MR | Zbl

[20] Zolotarev V. M., Senatov V. V., “Dvustoronnie otsenki matritsy Levi”, Teoriya veroyatn. i ee primen., 20:2 (1975), 239–250 | MR | Zbl

[21] Shiganov I. S., “Nekotorye otsenki, svyazannye s ustoichivostyu teoremy G. Kramera”, Zap. nauch. sem. LOMI, 87, 1979, 187–195 | MR | Zbl

[22] Shiganov I. S., “On stability estimates of Cramér's theorem”, Lecture Notes in Math., 1233, 1987, 178–181 | DOI | MR

[23] Zolotarev V. M., “K voprosu ob ustoichivosti razlozheniya normalnogo zakona raspredeleniya na komponenty”, Teoriya veroyatn. i ee primen., 13:4 (1968), 738–742 | MR

[24] Zolotarev V. M., “Neskolko novykh veroyatnostnykh neravenstv, svyazannykh s metrikoi Levi”, Dokl. AN SSSR, 190:5 (1970), 1019–1021 | MR | Zbl

[25] Zolotarev V. M., “Otsenki razlichiya raspredelenii v metrike Levi”, Tr. MIAN SSSR, 112, 1971, 224–231 | MR | Zbl