We prove a functional central limit theorem for the empirical process of a stationary process X t =Y t +V t , where Y t is a long memory moving average in i.i.d. r.v.’s ζ s , s ≤ t, and V t =V (ζ t , ζt-1,...) is a weakly dependent nonlinear Bernoulli shift. Conditions of weak dependence of V t are written in terms of L2-norms of shift-cut differences V (ζ t , ζt-n, 0,...,) − V(ζ t ,...,ζt-n+1, 0,...). Examples of Bernoulli shifts are discussed. The limit empirical process is a degenerated process of the form f(x)Z, where f is the marginal p.d.f. of X0 and Z is a standard normal r.v. The proof is based on a uniform reduction principle for the empirical process.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Dehling M. S. Taqqu (1989) ArticleTitleThe empirical process of some long-range dependent sequences with an application to U-statistics Ann. Stat. 17 1767–1783
P. Diaconis D. Freedman (1999) ArticleTitleIterated random functions SIAM Review 41 45–76
R. L. Dobrushin P. Major (1979) ArticleTitleNon-central limit theorems for non-linear functions of Gaussian fields Z. Wahrsch. verw. Geb. 50 27–52
P Doukhan (1994) Mixing: Properties and Examples. Lecture Notes in Statistics, Vol. 85 Springer-Verlag New York
Doukhan, P. (2002). Models, inequalities and limit theorems for stationary sequence. In Taqqu, M. S., Oppenheim, G., Doukhan, P. (Eds.), Long-range dependence, theory and applications, Birkhaüser.
P. Doukhan G. Lang D. Surgailis (2002) ArticleTitleAsymptotics of weighted empirical processes of linear fields with long-range dependence Ann. Inst. Henri Poincaré Probabilités et Statistiques 38 IssueID6 879–896
M. Duflo (1990) Méthodes Récursives Aléatoires (English Edition, Springer-Verlag, 1996) Masson, Paris
L. Giraitis P. Kokoszka R. Leipus (2000) ArticleTitleStationary ARCH models: dependence structure and CLT Econometric Theory 16 3–22
L. Giraitis H. L. Koul D. Surgailis (1996) ArticleTitleAsymptotic normality of regression estimators with long memory errors Statist. Probab. Lett. 29 317–335
L. Giraitis D. Surgailis (1999) ArticleTitleCentral limit theorem for the empirical process of a linear sequence with long memory J. Stat. Plan. Inf. 80 81–93
L. Giraitis D. Surgailis (2002) ArticleTitleARCH-type bilinear models with double long memory Stoch. Process. Appl. 100 275–300
H.-C. Ho T. Hsing (1996) ArticleTitleOn the asymptotic expansion of the empirical process of long memory moving averages Ann. Statist. 24 992–1024
O Kallenberg (1997) Foundations of Modern Probability Springer-Verlag New-York
P. M. Robinson (1991) ArticleTitleTesting for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression J. Econometrics 47 67–84
Rosenblatt, M. (1961). Independence and dependence. In: Proc. Fourth Berkeley Symp. Math. Statist. Probab., Univ. California Press, Berkeley. pp. 411–443
D. Surgailis (2000) ArticleTitleLong-range dependence and Appell rank Ann. Probab. 28 478–497
M. S. Taqqu (1975) ArticleTitleWeak convergence to fractional Brownian motion and to the Rosenblatt process Z. Wahrsch. verw. Geb. 31 287–302
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Doukhan, P., Lang, G., Surgailis, D. et al. Functional Limit Theorem for the Empirical Process of a Class of Bernoulli Shifts with Long Memory. J Theor Probab 18, 161–186 (2005). https://doi.org/10.1007/s10959-004-2593-3
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s10959-004-2593-3