# Asymptotic behavior of the empirical process for gaussian data presenting seasonal long-memory

ESAIM: Probability and Statistics (2002)

- Volume: 6, page 293-309
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topHaye, Mohamedou Ould. "Asymptotic behavior of the empirical process for gaussian data presenting seasonal long-memory." ESAIM: Probability and Statistics 6 (2002): 293-309. <http://eudml.org/doc/245390>.

@article{Haye2002,

abstract = {We study the asymptotic behavior of the empirical process when the underlying data are gaussian and exhibit seasonal long-memory. We prove that the limiting process can be quite different from the limit obtained in the case of regular long-memory. However, in both cases, the limiting process is degenerated. We apply our results to von–Mises functionals and $U$-Statistics.},

author = {Haye, Mohamedou Ould},

journal = {ESAIM: Probability and Statistics},

keywords = {empirical process; Hermite polynomials; Rosenblatt processes; seasonal long-memory; $U$-statistics; von–Mises functionals},

language = {eng},

pages = {293-309},

publisher = {EDP-Sciences},

title = {Asymptotic behavior of the empirical process for gaussian data presenting seasonal long-memory},

url = {http://eudml.org/doc/245390},

volume = {6},

year = {2002},

}

TY - JOUR

AU - Haye, Mohamedou Ould

TI - Asymptotic behavior of the empirical process for gaussian data presenting seasonal long-memory

JO - ESAIM: Probability and Statistics

PY - 2002

PB - EDP-Sciences

VL - 6

SP - 293

EP - 309

AB - We study the asymptotic behavior of the empirical process when the underlying data are gaussian and exhibit seasonal long-memory. We prove that the limiting process can be quite different from the limit obtained in the case of regular long-memory. However, in both cases, the limiting process is degenerated. We apply our results to von–Mises functionals and $U$-Statistics.

LA - eng

KW - empirical process; Hermite polynomials; Rosenblatt processes; seasonal long-memory; $U$-statistics; von–Mises functionals

UR - http://eudml.org/doc/245390

ER -

## References

top- [1] M.A. Arcones, Distributional limit theorems over a stationary Gaussian sequence of random vectors. Stochastic Process. Appl. 88 (2000) 135-159. Zbl1045.60019MR1761993
- [2] 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). Zbl1031.65010
- [3] P. Billingsley, Convergence of Probability measures. Wiley (1968). Zbl0172.21201MR233396
- [4] S. Csörgo and J. Mielniczuk, The empirical process of a short-range dependent stationary sequence under Gaussian subordination. Probab. Theory Related Fields 104 (1996) 15-25. Zbl0838.60030MR1367664
- [5] H. Dehling and M.S. Taqqu, The empirical process of some long-range dependent sequences with an application to $U$-statistics. Ann. Statist. 4 (1989) 1767-1783. Zbl0696.60032MR1026312
- [6] H. Dehling and M.S. Taqqu, Bivariate symmetric statistics of long-range dependent observations. J. Statist. Plann. Inference 28 (1991) 153-165. Zbl0737.62042MR1115815
- [7] R.L. Dobrushin and P. Major, Non central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrsch. Verw. Geb. 50 (1979) 27-52. Zbl0397.60034MR550122
- [8] P. Doukhan and S. Louhichi, A new weak dependence condition and applications to moment inequalities. Stochastic Process Appl. 84 (1999) 313-342. Zbl0996.60020MR1719345
- [9] P. Doukhan and D. Surgailis, 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. Zbl0948.60012MR1649521
- [10] J. Ghosh, A new graphical tool to detect non normality. J. Roy. Statist. Soc. Ser. B 58 (1996) 691-702. Zbl0860.62004MR1410184
- [11] L. Giraitis, Convergence of certain nonlinear transformations of a Gaussian sequence to self-similar process. Lithuanian Math. J. 23 (1983) 58-68. Zbl0526.60031MR705726
- [12] L. Giraitis and R. Leipus, A generalized fractionally differencing approach in long-memory modeling. Lithuanian Math. J. 35 (1995) 65-81. Zbl0837.62066MR1357812
- [13] 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. Zbl0943.60035MR1713796
- [14] I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products. Jeffrey A. 5th Edition. Academic Press (1994). Zbl0918.65002MR1243179
- [15] H.C. Ho and T. Hsing, On the asymptotic expansion of the empirical process of long memory moving averages. Ann. Statist. 24 (1996) 992-1024. Zbl0862.60026MR1401834
- [16] I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus. Springer-Verlag, New York (1988). Zbl0638.60065MR917065
- [17] R. Leipus and M.-C. Viano, Modeling long-memory time series with finite or infinite variance: A general approach. J. Time Ser. Anal. 21 (1997) 61-74. Zbl0974.62083MR1766174
- [18] C. Newman, Asymptotic independence and limit theorems for positively and negatively dependent random variables. IMS Lecture Notes-Monographs Ser. 5 (1984) 127-140. MR789244
- [19] G. Oppenheim, M. Ould Haye and M.-C. Viano, Long memory with seasonal effects. Statist. Inf. Stoch. Proc. 3 (2000) 53-68. Zbl0979.60011MR1819286
- [20] M. Ould Haye, Longue mémoire saisonnière et convergence vers le processus de Rosenblatt. Pub. IRMA, Lille, 50-VIII (1999).
- [21] M. Ould Haye, Asymptotic behavior of the empirical process for seasonal long-memory data. Pub. IRMA, Lille, 53-V (2000).
- [22] 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). Zbl1033.60028MR1956045
- [23] D.W. Pollard, Convergence of Stochastic Processes. Springer, New York (1984). Zbl0544.60045MR762984
- [24] M. Rosenblatt, Limit theorems for transformations of functionals of Gaussian sequences. Z. Wahrsch. Verw. Geb. 55 (1981) 123-132. Zbl0447.60016MR608012
- [25] Q. Shao and H. Yu, Weak convergence for weighted empirical process of dependent sequences. Ann. Probab. 24 (1996) 2094-2127. Zbl0874.60006MR1415243
- [26] G.R. Shorack and J.A. Wellner, Empirical Processes with Applications to Statistics. Wiley, New York (1986). Zbl1170.62365MR838963
- [27] M.S. Taqqu, Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Geb. 31 (1975) 287-302. Zbl0303.60033MR400329
- [28] A. Zygmund, Trigonometric Series. Cambridge University Press (1959). Zbl0085.05601MR107776

## NotesEmbed ?

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