# Empirical processes of multidimensional systems with multiple mixing properties

@article{Dehling2010EmpiricalPO, title={Empirical processes of multidimensional systems with multiple mixing properties}, author={Herold Dehling and Olivier Durieu}, journal={Stochastic Processes and their Applications}, year={2010}, volume={121}, pages={1076-1096} }

We establish a multivariate empirical process central limit theorem for stationary -valued stochastic processes (Xi)i>=1 under very weak conditions concerning the dependence structure of the process. As an application, we can prove the empirical process CLT for ergodic torus automorphisms. Our results also apply to Markov chains and dynamical systems having a spectral gap on some Banach space of functions. Our proof uses a multivariate extension of the techniques introduced by Dehling etÂ al…

## 15 Citations

A sequential empirical CLT for multiple mixing processes with application toB-geometrically ergodic Markov chains

- Mathematics
- 2014

We investigate the convergence in distribution of sequential empirical processes of dependent data indexed by a class of functions F. Our technique is suitable for processes that satisfy a multiple…

A sequential empirical CLT for multiple mixing processes with application to $\mathcal{B}$-geometrically ergodic Markov chains

- Mathematics
- 2014

We investigate the convergence in distribution of sequential empirical processes of dependent data indexed by a class of functions F. Our technique is suitable for processes that satisfy a multiple…

Empirical processes of Markov chains and dynamical systems indexed by classes of functions

- Computer Science
- 2012

Weak convergence of empirical processes of dependent data, indexed by classes of functions, are studied, especially suitable for data arising from dynamical systems and Markov chains, where the Central Limit Theorem for partial sums is commonly derived via the spectral gap technique.

2 Empirical central limit theorems 2 . 1 Empirical central limit theorem in L

- 2021

Let T be an ergodic automorphism of the d-dimensional torus T, and f be a continuous function from T to R. On the probability space T equipped with the Lebesgue-Haar measure, we prove the weak…

Empirical central limit theorems for ergodic automorphisms of the torus

- Mathematics
- 2012

Let T be an ergodic automorphism of the d-dimensional torus T^d, and f be a continuous function from T^d to R^l. On the probability space T^d equipped with the Lebesgue-Haar measure, we prove the…

New techniques for empirical processes of dependent data

- Mathematics
- 2008

We present a new technique for proving the empirical process invariance principle for stationary processes (Xn)n>=0. The main novelty of our approach lies in the fact that we only require the central…

On the invariance principle for empirical processes of associated sequences

- Mathematics
- 2015

We consider empirical processes generated by strictly stationary sequences of associated random variables. S. Louhichi established an invariance principle for such processes, assuming that the…

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

- Mathematics
- 2014

We study weak convergence of empirical processes of dependent data $(X_i)_{i\geq 0}$, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and…

Empirical processes of iterated maps that contract on average

- Mathematics
- 2012

We consider a Markov chain obtained by random iterations of Lipschitz maps Ti chosen with a probability pi(x) depending on the current position x. We assume this system has a property of “contraction…

An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

- Mathematics
- 2014

Let $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ be the empirical process associated to an ℝd-valued stationary process (Xi)i≥0. In the present paper, we introduce very general conditions for weak convergence…

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