• Bernoulli
  • Volume 20, Number 3 (2014), 1372-1403.

Approximating class approach for empirical processes of dependent sequences indexed by functions

Herold Dehling, Olivier Durieu, and Marco Tusche

Full-text: Open access


We study weak convergence of empirical processes of dependent data $(X_{i})_{i\geq0}$, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique. We are specifically interested in situations where the index class $\mathcal{F}$ is different from the class of functions $f$ for which we have good properties of the observables $(f(X_{i}))_{i\geq0}$. We introduce a new bracketing number to measure the size of the index class $\mathcal{F}$ which fits this setting. Our results apply to the empirical process of data $(X_{i})_{i\geq0}$ satisfying a multiple mixing condition. This includes dynamical systems and Markov chains, if the Perron–Frobenius operator or the Markov operator has a spectral gap, but also extends beyond this class, for example, to ergodic torus automorphisms.

Article information

Bernoulli, Volume 20, Number 3 (2014), 1372-1403.

First available in Project Euclid: 11 June 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Empirical processes indexed by classes of functions dependent data Markov chains dynamical systems ergodic torus automorphism weak convergence


Dehling, Herold; Durieu, Olivier; Tusche, Marco. Approximating class approach for empirical processes of dependent sequences indexed by functions. Bernoulli 20 (2014), no. 3, 1372--1403. doi:10.3150/13-BEJ525.

Export citation


  • [1] Andrews, D.W. and Pollard, D. (1994). An introduction to functional central limit theorems for dependent stochastic processes. International Statistical Review 62 119–132.
  • [2] Beutner, E. and Zähle, H. (2012). Deriving the asymptotic distribution of U- and V-statistics of dependent data using weighted empirical processes. Bernoulli 18 803–822.
  • [3] Bickel, P.J. and Wichura, M.J. (1971). Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 1656–1670.
  • [4] Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.
  • [5] 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.
  • [6] Bradley, R.C. (2007). Introduction to Strong Mixing Conditions. Vols 13. Heber City, UT: Kendrick Press.
  • [7] Collet, P., Martinez, S. and Schmitt, B. (2004). Asymptotic distribution of tests for expanding maps of the interval. Ergodic Theory Dynam. Systems 24 707–722.
  • [8] 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. New York: Springer.
  • [9] Dedecker, J. and Prieur, C. (2007). An empirical central limit theorem for dependent sequences. Stochastic Process. Appl. 117 121–142.
  • [10] Dehling, H. and Durieu, O. (2011). Empirical processes of multidimensional systems with multiple mixing properties. Stochastic Process. Appl. 121 1076–1096.
  • [11] Dehling, H., Durieu, O. and Volný, D. (2009). New techniques for empirical processes of dependent data. Stochastic Process. Appl. 119 3699–3718.
  • [12] Dehling, H. and Philipp, W. (2002). Empirical process techniques for dependent data. In Empirical Process Techniques for Dependent Data 3–113. Boston, MA: Birkhäuser.
  • [13] Dhompongsa, S. (1984). A note on the almost sure approximation of the empirical process of weakly dependent random vectors. Yokohama Math. J. 32 113–121.
  • [14] Dolgopyat, D. (2004). Limit theorems for partially hyperbolic systems. Trans. Amer. Math. Soc. 356 1637–1689 (electronic).
  • [15] Donsker, M.D. (1952). Justification and extension of Doob’s heuristic approach to the Komogorov–Smirnov theorems. Ann. Math. Statist. 23 277–281.
  • [16] Doob, J.L. (1949). Heuristic approach to the Kolmogorov–Smirnov theorems. Ann. Math. Statist. 20 393–403.
  • [17] Doukhan, P. and Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stochastic Process. Appl. 84 313–342.
  • [18] Doukhan, P., Massart, P. and Rio, E. (1995). Invariance principles for absolutely regular empirical processes. Ann. Inst. Henri Poincaré Probab. Stat. 31 393–427.
  • [19] Dudley, R.M. (1966). Weak convergences of probabilities on nonseparable metric spaces and empirical measures on Euclidean spaces. Illinois J. Math. 10 109–126.
  • [20] Dudley, R.M. (1978). Central limit theorems for empirical measures. Ann. Probab. 6 899–929.
  • [21] Durieu, O. (2008). A fourth moment inequality for functionals of stationary processes. J. Appl. Probab. 45 1086–1096.
  • [22] Durieu, O. and Jouan, P. (2008). Empirical invariance principle for ergodic torus automorphisms; genericity. Stoch. Dyn. 8 173–195.
  • [23] Durieu, O. and Tusche, M. (2012). An empirical process central limit theorem for multidimensional dependent data. J. Theoret. Probab. DOI:10.1007/s10959-012-0450-3.
  • [24] 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. Berlin: Springer.
  • [25] Le Borgne, S. (1999). Limit theorems for non-hyperbolic automorphisms of the torus. Israel J. Math. 109 61–73.
  • [26] Leonov, V.P. (1960). On the central limit theorem for ergodic endomorphisms of compact commutative groups. Dokl. Akad. Nauk SSSR 135 258–261.
  • [27] Neuhaus, G. (1971). On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Statist. 42 1285–1295.
  • [28] Nolan, D. and Pollard, D. (1987). $U$-processes: Rates of convergence. Ann. Statist. 15 780–799.
  • [29] Ossiander, M. (1987). A central limit theorem under metric entropy with $L_{2}$ bracketing. Ann. Probab. 15 897–919.
  • [30] Philipp, W. (1984). Invariance principles for sums of mixing random elements and the multivariate empirical process. In Limit Theorems in Probability and Statistics, Vols I, II (Veszprém, 1982). Colloquia Mathematica Societatis János Bolyai 36 843–873. Amsterdam: North-Holland.
  • [31] Philipp, W. and Pinzur, L. (1980). Almost sure approximation theorems for the multivariate empirical process. Z. Wahrsch. Verw. Gebiete 54 1–13.
  • [32] Rio, E. (1998). Processus empiriques absolument réguliers et entropie universelle. Probab. Theory Related Fields 111 585–608.
  • [33] Straf, M.L. (1972). Weak convergence of stochastic processes with several parameters. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability Theory 187–221. Berkeley, CA: Univ. California Press.
  • [34] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. New York: Springer.