Geodesic
A Gibbs conditional theorem under extreme deviation
Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 3, pp. 489-518 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We explore some properties of the conditional distribution of an independently and identically distributed (i.i.d.) sample under large exceedances of its sum. Thresholds for the asymptotic independence of the summands are observed, in contrast with the classical case when the conditioning event is in the range of a large deviation. This paper is an extension of Broniatowski and Cao [Extremes, 17 (2014), pp. 305–336]. Tools include a new Edgeworth expansion adapted to specific triangular arrays, where the rows are generated by tilted distribution with diverging parameters, and some Abelian type results.
Keywords: Gibbs conditional principle, extreme deviation. 
@article{TVP_2022_67_3_a4,
     author = {M. Biret and M. Broniatowski and Z. Cao},
     title = {A~Gibbs conditional theorem under extreme deviation},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {489--518},
     year = {2022},
     volume = {67},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2022_67_3_a4/}
}
TY  - JOUR
AU  - M. Biret
AU  - M. Broniatowski
AU  - Z. Cao
TI  - A Gibbs conditional theorem under extreme deviation
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2022
SP  - 489
EP  - 518
VL  - 67
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TVP_2022_67_3_a4/
LA  - ru
ID  - TVP_2022_67_3_a4
ER  - 
%0 Journal Article
%A M. Biret
%A M. Broniatowski
%A Z. Cao
%T A Gibbs conditional theorem under extreme deviation
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2022
%P 489-518
%V 67
%N 3
%U http://geodesic.mathdoc.fr/item/TVP_2022_67_3_a4/
%G ru
%F TVP_2022_67_3_a4
M. Biret; M. Broniatowski; Z. Cao. A Gibbs conditional theorem under extreme deviation. Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 3, pp. 489-518. http://geodesic.mathdoc.fr/item/TVP_2022_67_3_a4/

[1] A. A. Balkema, C. Klüppelberg, S. I. Resnick, “Densities with Gaussian tails”, Proc. London Math. Soc. (3), 66:3 (1993), 568–588 | DOI | MR | Zbl

[2] S. K. Bar-Lev, D. Bshouty, P. Enis, “On polynomial variance functions”, Probab. Theory Related Fields, 94:1 (1992), 69–82 | DOI | MR | Zbl

[3] O. Barndorff-Nielsen, Information and exponential families in statistical theory, Wiley Ser. Probab. Math. Statist., John Wiley Sons, Ltd., Chichester, 1978, ix+238 pp. | DOI | MR | Zbl

[4] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular variation, Encyclopedia Math. Appl., 27, Cambridge Univ. Press, Cambridge, 1987, xx+491 pp. | DOI | MR | Zbl

[5] M. Biret, M. Broniatowski, Zansheng Cao, “A sharp Abelian theorem for the Laplace transform”, Mathematical statistics and limit theorems, Springer, Cham, 2015, 67–92 | DOI | MR | Zbl

[6] M. Broniatowski, A. Fuchs, “Tauberian theorems, Chernoff inequality, and the tail behavior of finite convolutions of distribution functions”, Adv. Math., 116:1 (1995), 12–33 | DOI | MR | Zbl

[7] M. Broniatowski, Zhansheng Cao, “Light tails: all summands are large when the empirical mean is large”, Extremes, 17:2 (2014), 305–336 | DOI | MR | Zbl

[8] M. Broniatowski, V. Caron, “Long runs under a conditional limit distribution”, Ann. Appl. Probab., 24:6 (2014), 2246–2296 | DOI | MR | Zbl

[9] I. Csiszár, “Sanov property, generalized $I$-projection and a conditional limit theorem”, Ann. Probab., 12:3 (1984), 768–793 | DOI | MR | Zbl

[10] A. Dembo, O. Zeitouni, “Refinements of the Gibbs conditioning principle”, Probab. Theory Related Fields, 104:1 (1996), 1–14 | DOI | MR | Zbl

[11] P. Diaconis, D. A. Freedman, “Conditional limit theorems for exponential families and finite versions of de Finetti's theorem”, J. Theoret. Probab., 1:4 (1988), 381–410 | DOI | MR | Zbl

[12] W. Feller, An introduction to probability theory and its applications, v. 2, 2nd ed., John Wiley Sons, Inc., New York–London–Sydney, 1971, xxiv+669 pp. | MR | MR | Zbl | Zbl

[13] U. Frisch, D. Sornette, “Extreme deviations and applications”, J. Physique I, 7:9 (1997), 1155–1171 | DOI

[14] J. L. Jensen, Saddlepoint approximations, Oxford Statist. Sci. Ser., 16, Oxford Univ. Press, New York, 1995, xii+332 pp. | MR | Zbl

[15] B. Jørgensen, J. R. Martínez, “Tauber theory for infinitely divisible variance functions”, Bernoulli, 3:2 (1997), 213–224 | DOI | MR | Zbl

[16] D. Juszczak, A. V. Nagaev, “Local large deviation theorem for sums of i.i.d. random vectors when the Cramér condition holds in the whole space”, Probab. Math. Statist., 24:2 (2004), 297–320 | MR | Zbl

[17] G. Letac, M. Mora, “Natural real exponential families with cubic variance functions”, Ann. Statist., 18:1 (1990), 1–37 | DOI | MR | Zbl

[18] D. Sornette, Critical phenomena in natural sciences. Chaos, fractals, selforganization and disorder: concepts and tools, Springer Ser. Synergetics, 2nd print. of the 2nd ed., Springer-Verlag, Berlin, 2006, xxii+528 pp. | DOI | MR | Zbl