Abelian theorems, limit properties of conjugate distributions,

A. Yu. Zaigraev   ; A. V. Nagaev

Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 4, pp. 701-719

Résumé

A class of multidimensional absolutely continuous distributions is considered. Each distribution has a moment generating function, which is finite in a bounded convex set $S$ and generates a family of the so-called conjugate distributions. We focus our attention on the limit distributions for this family when the conjugate parameter tends to the boundary of $S$. As in the one-dimensional case, each limit distribution is obtained as a corollary of the Abel-type theorem. The results obtained are utilized for establishing a local limit theorem for large deviations of arbitrarily high order.

Détail

Mots-clés : Cramér's condition
Keywords: deviation function, gamma-like distribution, large deviations of arbitrarily high order, local limit theorem, regular variation, support function.

Bibliographie

[1] Aleshkyavichene A. K., “Mnogomernye integralnye predelnye teoremy dlya veroyatnostei bolshikh uklonenii”, Teoriya veroyatn. i ee primen., 28:1 (1983), 62–82 | MR | Zbl

[2] von Bahr B., “Multi-dimensional integral limit theorems for large deviations”, Ark. Mat., 7 (1967), 89–99 | DOI | MR | Zbl

[3] Borovkov A. A., Rogozin B. A., “O tsentralnoi predelnoi teoreme v mnogomernom sluchae”, Teoriya veroyatn. i ee primen., 10:1 (1965), 61–69 | MR | Zbl

[4] Borovkov A. A., Mogulskii A. A., “Bolshie ukloneniya i proverka statisticheskikh gipotez. I: Bolshie ukloneniya summ sluchainykh vektorov”, Trudy In-ta matematiki SO RAN, 3, 1992, 1–63 | Zbl

[5] Bkhattachariya R. N., Ranga Rao R., Approksimatsiya normalnym raspredeleniem i asimptoticheskie razlozheniya, Nauka, M., 1982, 288 pp. | MR

[6] Dembo A., Zeitouni O., Large Deviations Techniques and Applications, Jones and Bartlett, Boston, 1993, 346 pp. | MR | Zbl

[7] Deuschel J.-D., Stroock D. W., Large Deviations, Academic Press, Boston, 1989, 307 pp. | MR | Zbl

[8] Ellis R. J., Entropy, Large Deviations and Statistical Mechanics, Springer-Verlag, New York, 1985, 364 pp. | MR

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

[10] Ibragimov I. A., Khasminskii R. Z., Asimptoticheskaya teoriya otsenivaniya, Nauka, M., 1979, 527 pp. | MR

[11] Lankaster P., Teoriya matrits, Nauka, M., 1978, 280 pp. | MR

[12] Nagaev A. V., “Bolshie ukloneniya dlya odnogo klassa raspredelenii”, Predelnye teoremy teorii veroyatnostei, Izd-vo AN UzSSR, Tashkent, 1963, 55–68

[13] Nagaev A. V., “Predelnye teoremy dlya summ nezavisimykh dvumernykh sluchainykh vektorov”, Predelnye teoremy i statisticheskie vyvody, Fan, Tashkent, 1966, 83–89 | MR

[14] Nagaev A. V., “Lokalnye predelnye teoremy spetsialnogo vida, uchityvayuschie bolshie ukloneniya”, Izv. AN UzSSR, ser. fiz.-matem., 14:1 (1970), 29–34 | MR | Zbl

[15] Nagaev A. V., Sakoyan S. K., “Predelnye teoremy, uchityvayuschie bolshie ukloneniya, v $\mathbb R^k$”, Dokl. AN SSSR, 204:3 (1972), 554–556 | MR | Zbl

[16] Nagaev A. V., Khodzhabagyan S. S., “Predelnye teoremy, uchityvayuschie bolshie ukloneniya dlya summ polozhitelnykh sluchainykh velichin”, Litov. matem. sb., 14:1 (1974), 149–163 | MR | Zbl

[17] Nagaev A. V., “Bolshie ukloneniya dlya summ reshetchatykh sluchainykh velichin pri vypolnenii usloviya Kramera”, Diskretn. matem., 10:3 (1998), 115–130 | MR | Zbl

[18] Nagaev A. V., “Kramerovskie bolshie ukloneniya v sluchae, kogda krainee sopryazhennoe raspredelenie imeet tyazhelyi khvost”, Teoriya veroyatn. i ee primen., 43:3 (1998), 456–475 | MR | Zbl

[19] Nagaev A. V., Zaigraev A., “Abelian theorems for a class of probability distributions in $\mathbb R^d$ and their application”, J. Math. Sci. (New York), 99:4 (2000), 1454–1462 | DOI | MR | Zbl

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

[21] Nagaev S. V., “Large deviations of sums of independent random variables”, Ann. Probab., 7:5 (1979), 745–789 | DOI | MR | Zbl

[22] Ney P., “Dominating points and the asymptotics of large deviations for random walk on $\mathbb R^d$”, Ann. Probab., 11:1 (1983), 158–167 | DOI | MR | Zbl

[23] Osipov L. V., “On large deviations for sums of random vectors in $\mathbb R^k$”, J. Multivariate Anal., 11:2 (1981), 115–126 | DOI | MR | Zbl

[24] Petrov V. V., “O veroyatnostyakh bolshikh uklonenii summ nezavisimykh sluchainykh velichin”, Teoriya veroyatn. i ee primen., 10:2 (1965), 310–322 | MR | Zbl

[25] Resnick S. I., Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York, 1987, 320 pp. | MR | Zbl

[26] Rikhter V., “Lokalnye predelnye teoremy dlya bolshikh uklonenii”, Dokl. AN SSSR, 115 (1957), 53–56

[27] Rozovsky L. V., “On probabilities of large deviations in some classes of $k$-dimensional Borel sets”, J. Multivariate Anal., 17:1 (1985), 1–26 | DOI | MR | Zbl

[28] Rozovskii L. V., “Veroyatnosti bolshikh uklonenii na vsei osi”, Teoriya veroyatn. i ee primen., 38:1 (1993), 79–109 | MR

[29] Rozovskii L. V., “Veroyatnosti bolshikh uklonenii summy nezavisimykh sluchainykh velichin s obschei funktsiei raspredeleniya iz oblasti prityazheniya nesimmetrichnogo ustoichivogo zakona”, Teoriya veroyatn. i ee primen., 42:3 (1997), 496–530 | MR

[30] Rokafellar R. T., Vypuklyi analiz, Mir, M., 1973, 468 pp.

[31] Saulis L., Statulyavichyus V., Predelnye teoremy o bolshikh ukloneniyakh, Mokslas, Vilnyus, 1989, 208 pp. | MR | Zbl

[32] Svetulyavichene V. K., “O veroyatnostyakh bolshikh uklonenii dlya summ sluchainykh vektorov”, Litov. matem. sb., 21:2 (1981), 191–199 | MR | Zbl