The Annals of Probability

Strong approximation results for the empirical process of stationary sequences

Jérôme Dedecker, Florence Merlevède, and Emmanuel Rio

Full-text: Open access

Abstract

We prove a strong approximation result for the empirical process associated to a stationary sequence of real-valued random variables, under dependence conditions involving only indicators of half lines. This strong approximation result also holds for the empirical process associated to iterates of expanding maps with a neutral fixed point at zero, as soon as the correlations decrease more rapidly than $n^{-1-\delta}$ for some positive $\delta$. This shows that our conditions are in some sense optimal.

Article information

Source
Ann. Probab., Volume 41, Number 5 (2013), 3658-3696.

Dates
First available in Project Euclid: 12 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1378991852

Digital Object Identifier
doi:10.1214/12-AOP798

Mathematical Reviews number (MathSciNet)
MR3127895

Zentralblatt MATH identifier
1284.60069

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60G10: Stationary processes 37E05: Maps of the interval (piecewise continuous, continuous, smooth)

Keywords
Strong approximation Kiefer process stationary sequences intermittent maps weak dependence

Citation

Dedecker, Jérôme; Merlevède, Florence; Rio, Emmanuel. Strong approximation results for the empirical process of stationary sequences. Ann. Probab. 41 (2013), no. 5, 3658--3696. doi:10.1214/12-AOP798. https://projecteuclid.org/euclid.aop/1378991852


Export citation

References

  • Berkes, I., Hörmann, S. and Schauer, J. (2009). Asymptotic results for the empirical process of stationary sequences. Stochastic Process. Appl. 119 1298–1324.
  • Berkes, I. and Philipp, W. (1977). An almost sure invariance principle for the empirical distribution function of mixing random variables. Z. Wahrsch. Verw. Gebiete 41 115–137.
  • Bickel, P. J. and Wichura, M. J. (1971). Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 1656–1670.
  • Borovkova, S., Burton, R. and Dehling, H. (2001). Limit theorems for functionals of mixing processes with applications to $U$-statistics and dimension estimation. Trans. Amer. Math. Soc. 353 4261–4318.
  • Castelle, N. and Laurent-Bonvalot, F. (1998). Strong approximations of bivariate uniform empirical processes. Ann. Inst. Henri Poincaré Probab. Stat. 34 425–480.
  • Dedecker, J. (2010). An empirical central limit theorem for intermittent maps. Probab. Theory Related Fields 148 177–195.
  • Dedecker, J., Gouëzel, S. and Merlevède, F. (2010). Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains. Ann. Inst. Henri Poincaré Probab. Stat. 46 796–821.
  • Dedecker, J. and Merlevède, F. (2010). On the almost sure invariance principle for stationary sequences of Hilbert-valued random variables. In Dependence in Probability, Analysis and Number Theory 157–175. Kendrick Press, Heber City, UT.
  • Dedecker, J., Prieur, C. and Raynaud De Fitte, P. (2006). Parametrized Kantorovich-Rubinštein theorem and application to the coupling of random variables. In Dependence in Probability and Statistics. Lecture Notes in Statistics 187 105–121. Springer, New York.
  • Dedecker, J. and Prieur, C. (2007). An empirical central limit theorem for dependent sequences. Stochastic Process. Appl. 117 121–142.
  • Dedecker, J. and Prieur, C. (2009). Some unbounded functions of intermittent maps for which the central limit theorem holds. ALEA Lat. Am. J. Probab. Math. Stat. 5 29–45.
  • Dedecker, J., Doukhan, P., Lang, G., León R., J. R., Louhichi, S. and Prieur, C. (2007). Weak Dependence: With Examples and Applications. Lecture Notes in Statistics 190. Springer, New York.
  • Dehling, H. and Taqqu, M. S. (1989). The empirical process of some long-range dependent sequences with an application to $U$-statistics. Ann. Statist. 17 1767–1783.
  • Dudley, R. M. and Philipp, W. (1983). Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrsch. Verw. Gebiete 62 509–552.
  • Finkelstein, H. (1971). The law of the iterated logarithm for empirical distributions. Ann. Math. Statist. 42 607–615.
  • Giraitis, L. and Surgailis, D. (2002). The reduction principle for the empirical process of a long memory linear process. In Empirical Process Techniques for Dependent Data 241–255. Birkhäuser, Boston, MA.
  • Hennion, H. and Hervé, L. (2001). Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Math. 1766. Springer, Berlin.
  • Kiefer, J. (1972). Skorohod embedding of multivariate rv’s, and the sample df. Z. Wahrsch. Verw. Gebiete 24 1–35.
  • Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent $\mathrm{RV}$’s and the sample $\mathrm{DF}$. I. Z. Wahrsch. Verw. Gebiete 32 111–131.
  • Lai, T. L. (1974). Reproducing kernel Hilbert spaces and the law of the iterated logarithm for Gaussian processes. Z. Wahrsch. Verw. Gebiete 29 7–19.
  • Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Ergebnisse der Mathematik und Ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 23. Springer, Berlin.
  • Liverani, C., Saussol, B. and Vaienti, S. (1999). A probabilistic approach to intermittency. Ergodic Theory Dynam. Systems 19 671–685.
  • Merlevède, F. and Rio, E. (2012). Strong approximation of partial sums under dependence conditions with application to dynamical systems. Stochastic Process. Appl. 122 386–417.
  • Rio, E. (2000). Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants. Mathématiques & Applications (Berlin) [Mathematics & Applications] 31. Springer, Berlin.
  • Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 43–47.
  • Rüschendorf, L. (1985). The Wasserstein distance and approximation theorems. Z. Wahrsch. Verw. Gebiete 70 117–129.
  • Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • Wu, W. B. (2007). Strong invariance principles for dependent random variables. Ann. Probab. 35 2294–2320.
  • Wu, W. B. (2008). Empirical processes of stationary sequences. Statist. Sinica 18 313–333.
  • Yoshihara, K.-i. (1979). Note on an almost sure invariance principle for some empirical processes. Yokohama Math. J. 27 105–110.
  • Yu, H. (1993). A Glivenko–Cantelli lemma and weak convergence for empirical processes of associated sequences. Probab. Theory Related Fields 95 357–370.
  • Zweimüller, R. (1998). Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points. Nonlinearity 11 1263–1276.