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Asymptotic behavior of the Empirical Process for Gaussian data presenting seasonallong-memory

Published online by Cambridge University Press:  15 November 2002

Mohamedou Ould Haye
Mohamedou Ould Haye*
Affiliation:
Laboratoire de Statistique et Probabilités, bâtiment M2, FRE 2222 du CNRS, Université des Sciences et Technologies de Lille, 59655 Villeneuve-d'Ascq Cedex, France; [email protected].
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Abstract

We study the asymptotic behavior of the empirical process when theunderlying data are Gaussian and exhibit seasonallong-memory. We prove that the limiting process can be quitedifferent from the limit obtained in the case of regularlong-memory. However, in both cases, the limiting process isdegenerated. We apply our results to von–Mises functionals andU-Statistics.

Keywords

Empirical processHermite polynomialsRosenblatt processesseasonal long-memoryU-statisticsvon–Mises functionals.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 6: New directions in Time Series Analysis (Guest Editor: Philippe Soulier)New directions in Time Series Analysis (Guest Editor: Philippe Soulier) , 2002 , pp. 293 - 309
Copyright
© EDP Sciences, SMAI, 2002

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References

Arcones, M.A., Distributional limit theorems over a stationary Gaussian sequence of random vectors. Stochastic Process. Appl. 88 (2000) 135-159. CrossRef
J.-M. Bardet, G. Lang, G. Oppenheim, A. Philippe and M.S. Taqqu, Generators of long-range processes: A survey, in Long range dependence: Theory and applications, edited by P. Doukhan, G. Oppenheim and M.S. Taqqu (to appear).
P. Billingsley, Convergence of Probability measures. Wiley (1968).
Csörgo, S. and Mielniczuk, J., The empirical process of a short-range dependent stationary sequence under Gaussian subordination. Probab. Theory Related Fields 104 (1996) 15-25. CrossRef
Dehling, H. and Taqqu, M.S., The empirical process of some long-range dependent sequences with an application to U-statistics. Ann. Statist. 4 (1989) 1767-1783. CrossRef
Dehling, H. and Taqqu, M.S., Bivariate symmetric statistics of long-range dependent observations. J. Statist. Plann. Inference 28 (1991) 153-165. CrossRef
Dobrushin, R.L. and Major, P., Non central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrsch. Verw. Geb. 50 (1979) 27-52. CrossRef
Doukhan, P. and Louhichi, S., A new weak dependence condition and applications to moment inequalities. Stochastic Process Appl. 84 (1999) 313-342. CrossRef
Doukhan, P. and Surgailis, D., Functional central limit theorem for the empirical process of short memory linear processes. C. R. Acad. Sci. Paris Sér. I Math. 326 (1997) 87-92. CrossRef
Ghosh, J., A new graphical tool to detect non normality. J. Roy. Statist. Soc. Ser. B 58 (1996) 691-702.
Giraitis, L., Convergence of certain nonlinear transformations of a Gaussian sequence to self-similar process. Lithuanian Math. J. 23 (1983) 58-68. CrossRef
Giraitis, L. and Leipus, R., A generalized fractionally differencing approach in long-memory modeling. Lithuanian Math. J. 35 (1995) 65-81. CrossRef
L. Giraitis and D. Surgailis Central limit theorem for the empirical process of a linear sequence with long memory. J. Statist. Plann. Inference 80 (1999) 81-93.
I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products. Jeffrey A. 5th Edition. Academic Press (1994).
Ho, H.C. and Hsing, T., On the asymptotic expansion of the empirical process of long memory moving averages. Ann. Statist. 24 (1996) 992-1024.
I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus. Springer-Verlag, New York (1988).
Leipus, R. and Viano, M.-C., Modeling long-memory time series with finite or infinite variance: A general approach. J. Time Ser. Anal. 21 (1997) 61-74. CrossRef
Newman, C., Asymptotic independence and limit theorems for positively and negatively dependent random variables. IMS Lecture Notes-Monographs Ser. 5 (1984) 127-140. CrossRef
Oppenheim, G., Ould Haye, M. and Viano, M.-C., Long memory with seasonal effects. Statist. Inf. Stoch. Proc. 3 (2000) 53-68. CrossRef
M. Ould Haye, Longue mémoire saisonnière et convergence vers le processus de Rosenblatt. Pub. IRMA, Lille, 50-VIII (1999).
M. Ould Haye, Asymptotic behavior of the empirical process for seasonal long-memory data. Pub. IRMA, Lille, 53-V (2000).
M. Ould Haye and M.-C. Viano, Limit theorems under seasonal long-memory, in Long range dependence: Theory and applications, edited by P. Doukhan, G. Oppenheim and M.S. Taqqu (to appear).
D.W. Pollard, Convergence of Stochastic Processes. Springer, New York (1984).
Rosenblatt, M., Limit theorems for transformations of functionals of Gaussian sequences. Z. Wahrsch. Verw. Geb. 55 (1981) 123-132. CrossRef
Shao, Q. and Weak, H. Yu convergence for weighted empirical process of dependent sequences. Ann. Probab. 24 (1996) 2094-2127. CrossRef
G.R. Shorack and J.A. Wellner, Empirical Processes with Applications to Statistics. Wiley, New York (1986).
Taqqu, M.S., Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Geb. 31 (1975) 287-302. CrossRef
A. Zygmund, Trigonometric Series. Cambridge University Press (1959).