Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences

Jérôme Dedecker; Sana Louhichi

ESAIM: Probability and Statistics (2005)

  • Volume: 9, page 38-73
  • ISSN: 1292-8100

Abstract

top
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.

How to cite

top

Dedecker, Jérôme, and Louhichi, Sana. "Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences." ESAIM: Probability and Statistics 9 (2005): 38-73. <http://eudml.org/doc/245269>.

@article{Dedecker2005,
abstract = {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.},
author = {Dedecker, Jérôme, Louhichi, Sana},
journal = {ESAIM: Probability and Statistics},
keywords = {infinitely divisible distributions; Lévy processes; weak dependence; association; binary random variables; number of exceedances; Infinitely divisible distributions},
language = {eng},
pages = {38-73},
publisher = {EDP-Sciences},
title = {Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences},
url = {http://eudml.org/doc/245269},
volume = {9},
year = {2005},
}

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
PB - EDP-Sciences
VL - 9
SP - 38
EP - 73
AB - 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.
LA - eng
KW - infinitely divisible distributions; Lévy processes; weak dependence; association; binary random variables; number of exceedances; Infinitely divisible distributions
UR - http://eudml.org/doc/245269
ER -

References

top
  1. [1] A. Araujo and E. Giné, The central limit theorem for real and Banach space valued random variables. Wiley, New York (1980). Zbl0457.60001MR576407
  2. [2] A.D. Barbour, L. Holst and S. Janson, Poisson approximation. Clarendon Press, Oxford (1992). Zbl0746.60002MR1163825
  3. [3] R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life: Probability Models. Silver Spring, MD (1981). Zbl0379.62080
  4. [4] T. Birkel, On the convergence rate in the central limit theorem for associated processes. Ann. Probab. 16 (1988) 1685–1698. Zbl0658.60039
  5. [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. Zbl0873.60011
  6. [6] L.H.Y. Chen, Poisson approximation for dependent trials. Ann. Probab. 3 (1975) 534–545. Zbl0335.60016
  7. [7] J.T. Cox and G. Grimmett, Central limit theorems for associated random variables and the percolation models. Ann. Probab. 12 (1984) 514–528. Zbl0536.60094
  8. [8] J. Dedecker and S. Louhichi, Conditional convergence to infinitely divisible distributions with finite variance. Stochastic Proc. Appl. (To appear.) Zbl1070.60033MR2132596
  9. [9] P. Doukhan and S. Louhichi, A new weak dependence condition and applications to moment inequalities. Stochastic Proc. Appl. 84 (1999) 313–342. Zbl0996.60020
  10. [10] J. Esary, F. Proschan and D. Walkup, Association of random variables with applications. Ann. Math. Statist. 38 (1967) 1466–1476. Zbl0183.21502
  11. [11] C. Fortuin, P. Kastelyn and J. Ginibre, Correlation inequalities on some ordered sets. Comm. Math. Phys. 22 (1971) 89–103. Zbl0346.06011
  12. [12] B.V. Gnedenko and A.N. Kolmogorov, Limit distributions for sums of independent random variables. Addison-Wesley Publishing Company (1954). Zbl0056.36001MR62975
  13. [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. Zbl0713.60047
  14. [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. Zbl0619.60054
  15. [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. Zbl0672.60033
  16. [16] A. Jakubowski, Minimal conditions in p -stable limit theorems. Stochastic Proc. Appl. 44 (1993) 291–327. Zbl0771.60015
  17. [17] A. Jakubowski, Minimal conditions in p -stable limit theorems -II. Stochastic Proc. Appl. 68 (1997) 1–20. Zbl0890.60024
  18. [18] K. Joag-Dev and F. Proschan, Negative association of random variables, with applications. Ann. Statist. 11 (1982) 286–295. Zbl0508.62041
  19. [19] O. Kallenberg, Random Measures. Akademie-Verlag, Berlin (1975). Zbl0345.60031MR431372
  20. [20] M. Kobus, Generalized Poisson Distributions as Limits of Sums for Arrays of Dependent Random Vectors. J. Multi. Analysis (1995) 199–244. Zbl0821.60032
  21. [21] M.R Leadbetter, G. Lindgren and H. Rootzén, Extremes and related properties of random sequences and processes. New York, Springer (1983). Zbl0518.60021MR691492
  22. [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. [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. Zbl0527.60050
  24. [24] C.M. Newman and A.L. Wright, An invariance principle for certain dependent sequences. Ann. Probab. 9 (1981) 671–675. Zbl0465.60009
  25. [25] V.V. Petrov, Limit theorems of probability theory: sequences of independent random variables. Clarendon Press, Oxford (1995). Zbl0826.60001MR1353441
  26. [26] L. Pitt, Positively Correlated Normal Variables are Associated. Ann. Probab. 10 (1982) 496–499. Zbl0482.62046
  27. [27] E. Rio, Théorie asymptotique des processus aléatoires faiblement dépendants. Collection Mathématiques & Applications. Springer, Berlin 31 (2000). Zbl0944.60008MR2117923
  28. [28] K.I. Sato, Lévy processes and infinitely divisible distributions. Cambridge studies in advanced mathematics 68 (1999). Zbl0973.60001MR1739520
  29. [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. Zbl0278.60026

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.