We continue the investigation started in a previous paper, on weak convergence to infinitely divisible distributions with finite variance. In the present paper, we study this problem for some weakly dependent random variables, including in particular associated sequences. We obtain minimal conditions expressed in terms of individual random variables. As in the i.i.d. case, we describe the convergence to the gaussian and the purely non-gaussian parts of the infinitely divisible limit. We also discuss the rate of Poisson convergence and emphasize the special case of Bernoulli random variables. The proofs are mainly based on Lindeberg's method.
Mots clés : infinitely divisible distributions, Lévy processes, weak dependence, association, binary random variables, number of exceedances
@article{PS_2005__9__38_0,
author = {Dedecker, J\'er\^ome and Louhichi, Sana},
title = {Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences},
journal = {ESAIM: Probability and Statistics},
pages = {38--73},
publisher = {EDP-Sciences},
volume = {9},
year = {2005},
doi = {10.1051/ps:2005003},
mrnumber = {2148960},
zbl = {1136.60308},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2005003/}
}
TY - JOUR AU - Dedecker, Jérôme AU - Louhichi, Sana TI - Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences JO - ESAIM: Probability and Statistics PY - 2005 SP - 38 EP - 73 VL - 9 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2005003/ DO - 10.1051/ps:2005003 LA - en ID - PS_2005__9__38_0 ER -
%0 Journal Article %A Dedecker, Jérôme %A Louhichi, Sana %T Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences %J ESAIM: Probability and Statistics %D 2005 %P 38-73 %V 9 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2005003/ %R 10.1051/ps:2005003 %G en %F PS_2005__9__38_0
Dedecker, Jérôme; Louhichi, Sana. Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 38-73. doi : 10.1051/ps:2005003. http://www.numdam.org/articles/10.1051/ps:2005003/
[1] and, The central limit theorem for real and Banach space valued random variables. Wiley, New York (1980). | MR
[2] , and, Poisson approximation. Clarendon Press, Oxford (1992). | MR | Zbl
[3] and, Statistical Theory of Reliability and Life: Probability Models. Silver Spring, MD (1981).
[4] , On the convergence rate in the central limit theorem for associated processes. Ann. Probab. 16 (1988) 1685-1698. | Zbl
[5] , On the convergence rates in the CLT for positively and negatively dependent random fields, in Probability Theory and Mathematical Statistics, I.A. Ibragimov and A. Yu. Zaitsev Eds. Gordon and Breach Publishers, Singapore, (1996) 3-14. | Zbl
[6] , Poisson approximation for dependent trials. Ann. Probab. 3 (1975) 534-545. | Zbl
[7] and, Central limit theorems for associated random variables and the percolation models. Ann. Probab. 12 (1984) 514-528. | Zbl
[8] and, Conditional convergence to infinitely divisible distributions with finite variance. Stochastic Proc. Appl. (To appear.) | MR | Zbl
[9] and, A new weak dependence condition and applications to moment inequalities. Stochastic Proc. Appl. 84 (1999) 313-342. | Zbl
[10] , and, Association of random variables with applications. Ann. Math. Statist. 38 (1967) 1466-1476. | Zbl
[11] , and, Correlation inequalities on some ordered sets. Comm. Math. Phys. 22 (1971) 89-103. | Zbl
[12] and, Limit distributions for sums of independent random variables. Addison-Wesley Publishing Company (1954). | MR | Zbl
[13] and, Poisson approximation using the Stein-Chen method and coupling: number of exceedances of Gaussian random variables. Ann. Probab. 18 (1990) 713-723. | Zbl
[14] , and, On the Exceedance Point Process for a Stationary Sequence. Probab. Theory Related Fields 78 (1988) 97-112. | Zbl
[15] , and, Limit distributions of sums of m-dependent Bernoulli random variables. Probab. Theory Related Fields 82 (1989) 9-17. | Zbl
[16] , Minimal conditions in -stable limit theorems. Stochastic Proc. Appl. 44 (1993) 291-327. | Zbl
[17] , Minimal conditions in -stable limit theorems -II. Stochastic Proc. Appl. 68 (1997) 1-20. | Zbl
[18] and, Negative association of random variables, with applications. Ann. Statist. 11 (1982) 286-295. | Zbl
[19] , Random Measures. Akademie-Verlag, Berlin (1975). | MR | Zbl
[20] , Generalized Poisson Distributions as Limits of Sums for Arrays of Dependent Random Vectors. J. Multi. Analysis (1995) 199-244. | Zbl
[21] , and, Extremes and related properties of random sequences and processes. New York, Springer (1983). | MR | Zbl
[22] , Asymptotic independence and limit theorems for positively and negatively dependent random variables, in Inequalities in Statistics and Probability, Y.L. Tong Ed. IMS Lecture Notes-Monograph Series 5 (1984) 127-140.
[23] , and, Nearest neighbors and voronoi regions in certain point processes. Adv. Appl. Prob. 15 (1983) 726-751. | Zbl
[24] and, An invariance principle for certain dependent sequences. Ann. Probab. 9 (1981) 671-675. | Zbl
[25] , Limit theorems of probability theory: sequences of independent random variables. Clarendon Press, Oxford (1995). | MR | Zbl
[26] , Positively Correlated Normal Variables are Associated. Ann. Probab. 10 (1982) 496-499. | Zbl
[27] , Théorie asymptotique des processus aléatoires faiblement dépendants 31 (2000). | MR | Zbl
[28] , Lévy processes and infinitely divisible distributions. Cambridge studies in advanced mathematics 68 (1999). | MR | Zbl
[29] , A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, in Proc. Sixth Berkeley Symp. Math. Statist. Probab. Univ. California Press 3 (1971) 583-602. | Zbl
Cité par Sources :
