Journal of Theoretical Probability

, Volume 27, Issue 1, pp 249–277 | Cite as

An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

  • Olivier Durieu
  • Marco TuscheEmail author


Let \((U_{n}(t))_{t\in\mathbb{R}^{d}}\) be the empirical process associated to an ℝ d -valued stationary process (X i ) i≥0. In the present paper, we introduce very general conditions for weak convergence of \((U_{n}(t))_{t\in\mathbb{R}^{d}}\), which only involve properties of processes (f(X i )) i≥0 for a restricted class of functions \(f\in\mathcal{G}\). Our results significantly improve those of Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011) and provide new applications.

The central interest in our approach is that it does not need the indicator functions which define the empirical process \((U_{n}(t))_{t\in\mathbb{R}^{d}}\) to belong to the class \(\mathcal{G}\). This is particularly useful when dealing with data arising from dynamical systems or functionals of Markov chains. In the proofs we make use of a new application of a chaining argument and generalize ideas first introduced in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011).

Finally we will show how our general conditions apply in the case of multiple mixing processes of polynomial decrease and causal functions of independent and identically distributed processes, which could not be treated by the preceding results in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011).


Multivariate empirical processes Limit theorems Multiple mixing Chaining 

Mathematics Subject Classification (2010)

62G30 60F17 60G10 



We would like to thank the anonymous referee for valuable comments and suggestions which helped to improve the first version of the paper. We are also grateful to Herold Dehling for many helpful discussions.

This research was partially supported by German Research Foundation grant DE 370-4 project New Techniques for Empirical Processes of Dependent Data.


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964)Université de ToursToursFrance
  2. 2.Fakultät für MathematikRuhr-Universität BochumBochumGermany

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