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Abstrakty
We prove the Marcinkiewicz-Zygmund SLLN (MZ-SLLN) of order p,p ∈[1,2[, for associated sequences, not necessarily stationary. Our assumption on the moment of the random variables is minimal. We present an example of an associated and strongly mixing sequence, with infinite variance, to which our results apply. The conditions yielding such results for this example are discussed.
Czasopismo
Rocznik
Tom
Strony
203--214
Opis fizyczny
Bibliogr. 18 poz.
Twórcy
autor
- Université de Paris-Sud, Laboratoire de mathématiques, Probabilités, statistique et modélisation, Bat. 425, 91405 Orsay cedex, France
Bibliografia
- [1] L. E. Baum and M. Katz, Convergence rates in the law of large numbers, Trans. Amer. Math. Soc. 120 (1965), pp. 108-123.
- [2] T. Birkel, A note on the strong law of large numbers for positively dependent random variables, Statist. Probab. Lett. 7 (1988), pp. 17-20.
- [3] K. C. Chanda and F. H. Ruymgaart, General linear processes: a property of the empirical process applied to density and mode estimation, J. Time Ser. Anal. 3 (11) (1990), pp. 185-199.
- [4] T. K. Chandra and S. Ghosal, Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables, Acta Math. Hungar. 4 (71) (1996), pp. 327-336.
- [5] A. R. Dąbrowski and A. Jakubowski, Stable limits for associated random variables, Ann. Probab. 1 (22) (1994), pp. 1-16. '
- [6] P. Doukhan, Mixing: Properties and Examples, Lecture Notes in Statist. 85, Springer, 1994.
- [7] J. Esary, F. Proschan and D. Walkup, Association of random variables with applications, Ann. Math. Statist. 38 (1967), pp. 1466-1476.
- [8] M. R. Leadbetter, G. Lindgren and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes, Springer, New York 1983.
- [9] S. Louhichi, Weak convergence for empirical processes of associated sequences, Université de Paris-Sud, Prépublication 98.36, 1998.
- [10] 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 Ser. 5 (1984), pp. 127-140.
- [11] C. M. Newman and A. L. Wright, An invariance principle for certain dependent sequences, Ann. Probab. 9 (1981), pp. 671-675.
- [12] T. D. Pham and L. T. Tr an, Some mixing properties of time series models, Stochastic Process. Appl. 19 (1985), pp. 297-303.
- [13] E. Rio, A maximal inequality and dependent Marcinkiewicz-Zygmund strong laws, Ann. Probab. 2 (23) (1995), pp. 918-937.
- [14] M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), pp. 43-47.
- [15] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes. Stochastic Modeling, New York-London 1994,
- [16] Q. M. Shao, Complete convergence for ос-mixing sequences, Statist. Probab. Lett 16 (1993), pp. 279-287.
- [17] Т. E. Wood, A Berry-Esseen theorem for associated random variables, Ann. Probab. 11 (1983), pp. 1042-1047.
- [18] H. Yu, A Glivenko-Cantelli lemma and weak convergence for empirical processes of associated. sequences, Probab. Theory Related Fields 95 (1993), pp. 357-370.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-bb64ff22-a6fc-46a2-a27e-da69de103806