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}
}
  • H. Dehling, O. Durieu
  • Published 7 April 2010
  • Mathematics
  • Stochastic Processes and their Applications
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 et al… 
A sequential empirical CLT for multiple mixing processes with application toB-geometrically ergodic Markov chains
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
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
TLDR
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
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
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
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
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
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
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
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
...
1
2
...

References

SHOWING 1-10 OF 26 REFERENCES
Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation
In this paper we develop a general approach for investigating the asymptotic distribution of functional Xn = f((Zn+k)k∈z) of absolutely regular stochastic processes (Zn)n∈z. Such functional occur
An empirical central limit theorem for dependent sequences
We prove a central limit theorem for the d-dimensional distribution function of a class of stationary sequences. The conditions are expressed in terms of some coefficients which measure the
Empirical Invariance Principle for Ergodic Torus Automorphisms. Genericity
We consider the dynamical system given by an algebraic ergodic automorphism T on a torus. We study a Central Limit Theorem for the empirical process associated to the stationary process (f◦Ti)i∈ℕ,
Limit theorems for iterated random functions
  • W. Wu, X. Shao
  • Mathematics
    Journal of Applied Probability
  • 2004
We study geometric moment contracting properties of nonlinear time series that are expressed in terms of iterated random functions. Under a Dini-continuity condition, a central limit theorem for
A new weak dependence condition and applications to moment inequalities
The purpose of this paper is to propose a unifying weak dependence condition. Mixing sequences, functions of associated or Gaussian sequences, Bernoulli shifts as well as models with a Markovian
New techniques for empirical processes of dependent data
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
New dependence coefficients. Examples and applications to statistics
Abstract.To measure the dependence between a real-valued random variable X and a σ-algebra , we consider four distances between the conditional distribution function of X given and the distribution
Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness
General Facts About The Method Purpose Of The Paper.- The Central Limit Theorems For Markov Chains Theorems A, B, C.- Quasi-Compact Operators of Diagonal Type And Their Perturbations.- First
Dynamical properties of quasihyperbolic toral automorphisms
We study the dynamical properties of ergodic toral autmorphisms that have some eigenvalues of modulus one. For such automorphisms, all sufficiently fine smooth partitions generate measurably, but
Weak Dependence: With Examples and Applications
This monograph is aimed at developing Doukhan/Louhichi's (1999) idea to measure asymptotic independence of a random process. The authors propose various examples of models fitting such conditions
...
1
2
3
...