On large and superlarge deviations of sums of independent random vectors under Cramér's condition. II

A. A. Borovkov   ; A. A. Mogul'skii

Teoriâ veroâtnostej i ee primeneniâ, Tome 51 (2006) no. 4, pp. 641-673

Résumé

The present paper is a continuation of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 51 (2007), pp. 227–255]. In that paper we studied, in the univariate case, the asymptotics of the probabilities that a sum of independent identically distributed random variables will hit a half-interval $[x,x+\Delta)$ in the zone of superlarge deviations when the relative (scaled) deviations $\alpha=x/n$ unboundedly increase together with the number of summands $n$ and, at the same time, remain in the analyticity domain of the large deviations rate function for the summands. In the multivariate case, the first part of the paper presented sufficient conditions which ensure that integrolocal and local theorems of the same universal type as in the large and normal deviations zones will also hold in the superlarge deviations zone. The second part of the paper deals with the same problems for three classes on the most wide-spread univariate distributions, for which one can obtain simple sufficient conditions, enabling one to find the asymptotics of the desired probabilities, as $x/n\to \infty$, in the above-mentioned universal form. These are the classes of the so-called exponentially and “superexponentially” fast decaying regular distributions. For them, we also establish limit theorems for the Cramér transforms with parameter values close to the “critical” one. Moreover, we obtain asymptotic expansion for the large deviations rate function.

Détail

Keywords: large deviations rate function, large deviations, integrolocal theorem, semi-exponential distributions, superexponential distributions, characterization of the normal distribution, limit theorems for Cramér transforms, asymptotic expansions of the large deviations rate function.
Mots-clés : superlarge deviations

Bibliographie

[1] Borovkov A. A., Mogulskii A. A., “O bolshikh i sverkhbolshikh ukloneniyakh summ nezavisimykh sluchainykh vektorov pri vypolnenii usloviya Kramera. I”, Teoriya veroyatn. i ee primen., 51:2 (2006), 260–294 | MR

[2] Borovkov A. A., Borovkov K. A., “O veroyatnostyakh bolshikh uklonenii dlya sluchainykh bluzhdanii. I. Raspredeleniya s pravilno menyayuschimisya khvostami”, Teoriya veroyatn. i ee primen., 46:2 (2001), 209–232 | MR | Zbl

[3] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, t. 2, Mir, M., 1967, 752 pp. | MR

[4] Borovkov A. A., Borovkov K. A., Asimptoticheskii analiz sluchainykh bluzhdanii. I. Medlenno ubyvayuschie raspredeleniya skachkov, Nauka, M. (to appear)

[5] Seneta E., Pravilno menyayuschiesya funktsii, Nauka, M., 1985, 141 pp. | MR | Zbl

[6] Bingham N. M., Goldie C. M., Teugels J. L., Regular Variation, Encyclopedia Math. Appl., 27, Cambridge Univ. Press, Cambridge, 1987, 491 pp. | MR | Zbl

[7] Nagaev A. V., “Lokalnye teoremy s uchetom bolshikh uklonenii”, Predelnye teoremy i veroyatnostnye protsessy, Fan, Tashkent, 1967, 71–88 | MR

[8] Nagaev A. V., “Predelnye teoremy dlya odnoi skhemy serii”, Predelnye teoremy i veroyatnostnye protsessy, Fan, Tashkent, 1967, 43–70 | MR

[9] Rvacheva E. L., “Ob oblastyakh prityazheniya mnogomernykh raspredelenii”, Uchenye zapiski Lvovskogo gos. un-ta, 6 (1958), 5–44

[10] Stone C., “A local limit theorem for nonlattice multi-dimensional distribution functions”, Ann. Math. Statist., 36 (1965), 546–551 | DOI | MR | Zbl

[11] Borovkov A. A., Mogulskii A. A., “Bolshie ukloneniya dlya tsepei Markova v polozhitelnom kvadrante”, Uspekhi matem. nauk, 56:5(341) (2001), 3–116 | MR | Zbl

[12] Rozovskii L. V., “O sverkhbolshikh ukloneniyakh summy nezavisimykh sluchainykh velichin s obschim absolyutno nepreryvnym raspredeleniem, udovletvoryayuschim usloviyu Kramera”, Teoriya veroyatn. i ee primen., 47:1 (2002), 78–103