Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences
ESAIM: Probability and Statistics, Tome 9 (2005), pp. 38-73.

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.

DOI : https://doi.org/10.1051/ps:2005003
Classification : 60E07,  60F05
Mots clés : infinitely divisible distributions, Lévy processes, weak dependence, association, binary random variables, number of exceedances
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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] A. Araujo and E. Giné, The central limit theorem for real and Banach space valued random variables. Wiley, New York (1980). | MR 576407

[2] A.D. Barbour, L. Holst and S. Janson, Poisson approximation. Clarendon Press, Oxford (1992). | MR 1163825 | Zbl 0746.60002

[3] R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life: Probability Models. Silver Spring, MD (1981).

[4] T. Birkel, On the convergence rate in the central limit theorem for associated processes. Ann. Probab. 16 (1988) 1685-1698. | Zbl 0658.60039

[5] A.V. Bulinski, 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 0873.60011

[6] L.H.Y. Chen, Poisson approximation for dependent trials. Ann. Probab. 3 (1975) 534-545. | Zbl 0335.60016

[7] J.T. Cox and G. Grimmett, Central limit theorems for associated random variables and the percolation models. Ann. Probab. 12 (1984) 514-528. | Zbl 0536.60094

[8] J. Dedecker and S. Louhichi, Conditional convergence to infinitely divisible distributions with finite variance. Stochastic Proc. Appl. (To appear.) | MR 2132596 | Zbl 1070.60033

[9] P. Doukhan and S. Louhichi, A new weak dependence condition and applications to moment inequalities. Stochastic Proc. Appl. 84 (1999) 313-342. | Zbl 0996.60020

[10] J. Esary, F. Proschan and D. Walkup, Association of random variables with applications. Ann. Math. Statist. 38 (1967) 1466-1476. | Zbl 0183.21502

[11] C. Fortuin, P. Kastelyn and J. Ginibre, Correlation inequalities on some ordered sets. Comm. Math. Phys. 22 (1971) 89-103. | Zbl 0346.06011

[12] B.V. Gnedenko and A.N. Kolmogorov, Limit distributions for sums of independent random variables. Addison-Wesley Publishing Company (1954). | MR 62975 | Zbl 0056.36001

[13] L. Holst and S. Janson, Poisson approximation using the Stein-Chen method and coupling: number of exceedances of Gaussian random variables. Ann. Probab. 18 (1990) 713-723. | Zbl 0713.60047

[14] T. Hsing, J. Hüsler and M.R. Leadbetter, On the Exceedance Point Process for a Stationary Sequence. Probab. Theory Related Fields 78 (1988) 97-112. | Zbl 0619.60054

[15] W.N. Hudson, H.G. Tucker and J.A Veeh, Limit distributions of sums of m-dependent Bernoulli random variables. Probab. Theory Related Fields 82 (1989) 9-17. | Zbl 0672.60033

[16] A. Jakubowski, Minimal conditions in p-stable limit theorems. Stochastic Proc. Appl. 44 (1993) 291-327. | Zbl 0771.60015

[17] A. Jakubowski, Minimal conditions in p-stable limit theorems -II. Stochastic Proc. Appl. 68 (1997) 1-20. | Zbl 0890.60024

[18] K. Joag-Dev and F. Proschan, Negative association of random variables, with applications. Ann. Statist. 11 (1982) 286-295. | Zbl 0508.62041

[19] O. Kallenberg, Random Measures. Akademie-Verlag, Berlin (1975). | MR 431372 | Zbl 0345.60031

[20] M. Kobus, Generalized Poisson Distributions as Limits of Sums for Arrays of Dependent Random Vectors. J. Multi. Analysis (1995) 199-244. | Zbl 0821.60032

[21] M.R Leadbetter, G. Lindgren and H. Rootzén, Extremes and related properties of random sequences and processes. New York, Springer (1983). | MR 691492 | Zbl 0518.60021

[22] C.M. Newman, 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] C.M. Newman, Y. Rinott and A. Tversky, Nearest neighbors and voronoi regions in certain point processes. Adv. Appl. Prob. 15 (1983) 726-751. | Zbl 0527.60050

[24] C.M. Newman and A.L. Wright, An invariance principle for certain dependent sequences. Ann. Probab. 9 (1981) 671-675. | Zbl 0465.60009

[25] V.V. Petrov, Limit theorems of probability theory: sequences of independent random variables. Clarendon Press, Oxford (1995). | MR 1353441 | Zbl 0826.60001

[26] L. Pitt, Positively Correlated Normal Variables are Associated. Ann. Probab. 10 (1982) 496-499. | Zbl 0482.62046

[27] E. Rio, Théorie asymptotique des processus aléatoires faiblement dépendants 31 (2000). | MR 2117923 | Zbl 0944.60008

[28] K.I. Sato, Lévy processes and infinitely divisible distributions. Cambridge studies in advanced mathematics 68 (1999). | MR 1739520 | Zbl 0973.60001

[29] C.M. Stein, 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 0278.60026

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