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

We study weak convergence of empirical processes of dependent data $(X_i)_{i\geq0}$, 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.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ525 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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

We study weak convergence of empirical processes of dependent data, indexed by classes of functions. We obtain results that are 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. Our results apply, e.g. to the empirical process of ergodic torus automorphisms

A Sequential Empirical Central Limit Theorem for Multiple Mixing Processes with Application to 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 mixing condition on a space of functions which differs from the class F. This situation occurs in the case of data arising from dynamical systems or Markov chains, for which the Perron--Frobenius or Markov operator, respectively, has a spectral gap on a restricted space. We provide applications to iterative Lipschitz models that contract on average.Comment: Also available on http://ejp.ejpecp.org/article/view/3216. Note that the content of this version is identical to the one publisheb by "Electronic Journal of Probability". However, due to the use of different LaTeX-classes, the page number may diffe

An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

Let $(U_n(t))_{t\in\R^d}$ be the empirical process associated to an $\R^d$-valued stationary process $(X_i)_{i\ge 0}$. We give general conditions, which only involve processes $(f(X_i))_{i\ge 0}$ for a restricted class of functions $f$, under which weak convergence of $(U_n(t))_{t\in\R^d}$ can be proved. This is particularly useful when dealing with data arising from dynamical systems or functional of Markov chains. This result improves those of [DDV09] and [DD11], where the technique was first introduced, and provides new applications.Comment: to appear in Journal of Theoretical Probabilit

Processus empiriques de données à mélange multiple

Cette thèse étudie la convergence en loi des processus empiriques de données à mélange multiple. Son contenu correspond aux articles : Durieu et Tusche (2012), Dehling, Durieu, et Tusche (2012), et Dehiing, Durieu et Tusche (2013). Nous suivons l’approche par approximation introduite dans Dehling, Durieu, et Vo1n (2009) et Dehling and Durieu (2011), qui ont établi des théorèmes limite centraux empiriques pour des variables aléatoires dépendants à valeurs dans R ou RAd, respectivement. En développant leurs techniques, nous généralisons leurs résultats à des espaces arbitraires et à des processus empiriques indexés par des classes de fonctions. De plus, nous étudions des processus empiriques séquentiels. Nos résultats s’appliquent aux chaînes de Markov B-géométriquement ergodiques, aux modèles itératifs lipschitziens, aux systèmes dynamiques présentant un trou spectral pour l’opérateur de Perron-Frobenius associé, ou encore, aux automorphismes du tore. Nous établissons des conditions garantissant la convergence du processus empirique de tels modèles vers un processus gaussien.The present thesis studies weak convergence of empirical processes of multiple mixing data. It is based on the articles Durieu and Tusche (2012), Dehling, Durieu, and Tusche (2012), and Dehling, Durieu, and Tusche (2013). We follow the approximating class approach introduced by Dehling, Durieu, and Voln (2009)and Dehling and Durieu (2011), who established empirical central limit theorems for dependent R- and R”d-valued random variables, respectively. Extending their technique, we generalize their results to arbitrary state spaces and to empirical processes indexed by classes of functions. Moreover we study sequential empirical processes. Our results apply to B-geometrically ergodic Markov chains, iterative Lipschitz models, dynamical systems with a spectral gap on the Perron—Frobenius operator, and ergodic toms automorphisms. We establish conditions under which the empirical process of such processes converges weakly to a Gaussian process

Processus empiriques de données à mélange multiple

Die vorliegende Arbeit basiert auf den Arbeiten Durieu und Tusche (2012), Dehling, Durieu und Tusche (2012) und Dehling, Durieu und Tusche (2013). Wir untersuchen die schwache Konvergenz des empirishcen Prozesses multipel mischender Zufallsvariablen. Dehling, Durieu und Volný (2009) entwickelten eine Approximationsklassen-Technik um empirische zentrale Grenzwertsätze abhängiger eindimensionaler und später multivariater Zufallsvariablen (Dehling und Durieu (2011)) zu beweisen. Unter Zuhilfenahme der Approximationklassen-Technik von dieser Technik erweitern wir ihre Ergebnisse auf funktionenklassenindizierte empirische und sequentielle empirische Prozesse abhängiger Zufallsvariablen in beliebigen Ereignisräumen. Unsere Grenzwertsätze können auf B-geometrisch ergodische Markov-Ketten, iterative Lipschitz Modelle, dynamische Systeme deren Perron-Frobenius Operator eine Spektrallücke aufweist und ergodische Torusautomorphismen angewandt werden.The present thesis studies weak convergence of empirical processes of multiple mixing data. It is based on the articles Durieu and Tusche (2012), Dehling, Durieu, and Tusche (2012), and Dehling, Durieu, and Tusche (2013). We follow the approximating class approach introduced by Dehling, Durieu, and Volný (2009) and Dehling and Durieu (2011), who established empirical central limit theorems for dependent R- and R$^{d}$-valued random variables, respectively. Extending their technique, we generalize their results to arbitrary state spaces and to empirical processes indexed by classes of functions. Moreover we study sequential empirical processes. Our results apply to B-geometrically ergodic Markov chains, iterative Lipschitz models, dynamical systems with a spectral gap on the Perron-Frobenius operator, and ergodic torus automorphisms. We establish conditions under which the empirical process of such processes converges weakly to a Gaussian process

An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

International audienceLet $(U_n(t))_{t\in\R^d}$ be the empirical process associated to an $\R^d$-valued stationary process $(X_i)_{i\ge 0}$. We give general conditions, which only involve processes $(f(X_i))_{i\ge 0}$ for a restricted class of functions $f$, under which weak convergence of $(U_n(t))_{t\in\R^d}$ can be proved. This is particularly useful when dealing with data arising from dynamical systems or functional of Markov chains. This result improves those of [DDV09] and [DD11], where the technique was first introduced, and provides new applications