Geodesic
On the Stein–Tikhomirov method and its applications in nonclassical limit theorems
Diskretnaya Matematika, Tome 19 (2007) no. 1, pp. 27-39 
Sh. K. Formanov. On the Stein–Tikhomirov method and its applications in nonclassical limit theorems. Diskretnaya Matematika, Tome 19 (2007) no. 1, pp. 27-39. http://geodesic.mathdoc.fr/item/DM_2007_19_1_a4/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we give necessary and sufficient conditions for validity of the central limit theorem for sums of independent random variables in terms of characteristic functions in a nonclassical setup where an essential contribution of individual variables to the sum is allowed. These conditions are formulated with the use of the Stein–Tikhomirov operator.

[1] Stein C., “A bound for the error in the normal approximation to the distribution of a sum of dependent random variables”, Proc. 6th Berkeley Symp. Math. Stat. Probab., v. 2, Univ. California Press, Berkeley, Calif., 1972, 583–602 | MR | Zbl

[2] Stein C., Approximation, computation of expectations, Inst. Math. Statist., Hayward, Calif., 1986 | MR

[3] Tikhomirov A. N., “O skorosti skhodimosti v tsentralnoi predelnoi teoreme dlya slabo zavisimykh velichin”, Teoriya veroyatnostei i ee primeneniya, 25:4 (1980), 800–818 | MR | Zbl

[4] Formanov Sh. K., “Metod Steina–Tikhomirova i neklassicheskaya tsentralnaya predelnaya teorema”, Doklady RAN, 376:4 (2001), 458–460 | MR | Zbl

[5] Petrovskii I. G., Lektsii po teorii obyknovennykh differentsialnykh uravnenii, Nauka, Moskva, 1970

[6] Zolotarev V. M., Sovremennaya teoriya summirovaniya nezavisimykh sluchainykh velichin, Nauka, Moskva, 1986

[7] Rotar V. I., “K obobscheniyu teoremy Lindeberga–Fellera”, Matem. zametki, 18:1 (1975), 129–135 | MR | Zbl

[8] Kiryanova L. V., Rotar V. I., “Otsenka skorosti skhodimosti v tsentralnoi predelnoi teoreme dlya martingalov”, Teoriya veroyatnostei i ee primeneniya, 36 (1991), 301–312 | MR

[9] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, Moskva, 1977