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In this paper, we give necessary and sufficient conditions for a stationary sequence of random variables with values in a separable Hilbert space to satisfy the conditional central limit theorem introduced in Dedecker and Merlevede (Ann. Probab. 30 (2002) 1044–1081). As a consequence, this theorem implies stable convergence of the normalized partial sums to a mixture of normal distributions. We also establish the functional version of this theorem. Next, we show that these conditions are satisfied for a large class of weakly dependent sequences, including strongly mixing sequences as well as mixingales. Finally, we present an application to linear processes generated by some stationary sequences of H -valued random variables.

Many popular robust estimators are $U$-quantiles, most notably the Hodges-Lehmann location estimator and the $Q_n$ scale estimator. We prove a functional central limit theorem for the sequential $U$-quantile process without any moment assumptions and under weak short-range dependence conditions. We further devise an estimator for the long-run variance and show its consistency, from which the convergence of the studentized version of the sequential $U$-quantile process to a standard Brownian motion follows. This result can be used to construct CUSUM-type change-point tests based on $U$-quantiles, which do not rely on bootstrapping procedures. We demonstrate this approach in detail at the example of the Hodges-Lehmann estimator for robustly detecting changes in the central location. A simulation study confirms the very good robustness and efficiency properties of the test. Two real-life data sets are analyzed.

In this paper we study the almost sure conditional central limit theorem in its functional form for a class of random variables satisfying a projective criterion. Applications to strongly mixing processes and non irreducible Markov chains are given. The proofs are based on the normal approximation of double indexed martingale-like sequences, a theory which has interest in itself., Comment: accepted for publication in AIHP

Weak invariance principles are established for strictly stationary weakly dependent sequences, having a decomposed strong mixing coefficient into two parts, one based on the strong mixing condition with a polynomial mixing rate and other based on the ρ-mixing condition. The result is applied to the output of the Tukey ‘3R smoother’.

We consider a parametric class Tγ of expanding maps of [0, 1] with a neutral fixed point at 0 for which there exists an unique invariant absolutely continuous probability measure νγ on [0, 1]. On the probability space ([0, 1], νγ), we prove the weak invariance principle for the partial sums of f○Tγi in some special cases involving non-standard normalization. We also prove new moment inequalities and exponential bounds for the partial sums of f○Tγi when f is some Hölder function such that f(0)=νγ(f).

The aim of this paper is to compare various criteria leading to the central limit theorem and the weak invariance principle. These criteria are the martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk SSSR 188 (1969), the projective criterion introduced by Dedecker in Probab. Theory Related Fields 110 (1998), which was subsequently improved by Dedecker and Rio in Ann. Inst. H. Poincar\'{e} Probab. Statist. 36 (2000) and the condition introduced by Maxwell and Woodroofe in Ann. Probab. 28 (2000) later improved upon by Peligrad and Utev in Ann. Probab. 33 (2005). We prove that in every ergodic dynamical system with positive entropy, if we consider two of these criteria, we can find a function in $\mathbb{L}^2$ satisfying the first but not the second., Comment: Published in at http://dx.doi.org/10.1214/07-AIHP123 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The aim of this thesis is the study of limit theorems for stationary sequences of random variables (in particular, from dynamical system).We concentrate on two results which are important by their applications in statistics. We first study the asymptotique behavior of sums of random variables, precisely the central limit theorem and its invariance principles. We also consider the invariance principle of empirical processes. For the Donsker's weak invariance principle, many results can be obtained by martingale approximations and more generaly by projective criteria.We compare four of these criteria and we show that they are independent of each other. These criteria are the martingale-coboundary decomposition (Gordin, 1969), the Hannan condition (1979), the Dedecker and Rio criterion (2000) and the Maxwell-Woodroofe condition (2000).Concerning the asympotic behavior of empirical processes, we establish an invariance principle in the case of toral automorphisms. This permits to generalize the known hyperbolic case and to find a first result for a partially hyperbolic transformation.We also propose a new approach, based on operator techniques, for establishing an invariance principle.This technique is well adapted to cases when we only have good properties for a class of functions not containing the indicators. In particular, this is the case for some dynamical systems for which the transfer operator admits a spectral gap.At the end, following a question by Burton and Denker (1987), we are interested in the class of processes for which the central limit theorem holds. To refer to the empirical processes case, we study in particular the sequences of partial sums of iterates of indicator functions., Cette thèse se consacre à l'étude de théorèmes limites pour des suites de variables aléatoires stationnaires (en particulier issues d'un système dynamique). Nous nous concentrons sur deux résultats importants, notamment par leurs applications en statistiques. Nous étudions tout d'abord le comportement limite des sommes de variables aléatoires, plus précisément le théorème limite central et son principe d'invariance. Ensuite nous considérons le principe d'invariance pour les processus empiriques.Dans le cadre du principe d'invariance faible de Donsker, plusieurs résultats s'obtiennent au travers d'approximations par des martingales et plus généralement par des critères projectifs. Nous comparons quatre de ces critères et montrons leur indépendance mutuelle. Les critères étudiés sont la décomposition martingale-cobord (Gordin, 1969), la condition de Hannan (1979), le critère de Dedecker et Rio (2000) etla condition de Maxwell et Woodroofe (2000).En ce qui concerne le comportement asymptotique des processus empiriques, nous établissons un principe d'invariance dans le cas des automorphismes du tore. Cela permet de sortir du cadre hyperbolique connu et d'obtenir un premier résultat pour une transformation partiellement hyperbolique.Nous proposons également une nouvelle approche, basée sur des méthodes d'opérateurs, permettant d'établir un principe d'invariance empirique. Cette méthode s'applique en particulier aux cas où l'on a de bonnes propriétés pour une classe de fonctions ne contenant pas les fonctions indicatrices. C'est en particulier le cas de certains systèmes dynamiques dont l'opérateur de transfert admet un trou spectral.En dernier lieu, suivant une question de Burton et Denker (1987), nous nous intéressons à la classe des processus pour lesquels le théorème limite central a lieu. En référence au cadre des processus empiriques, nous étudions en particulier les suites de sommes partielles des itérées d'une fonction indicatrice.

This thesis consists of four papers, all concerning random iteration of isometries. The papers are: I. Ambroladze A, Ådahl M, Random iteration of isometries in unbounded metric spaces. Nonlinearity 16 (2003) 1107-1117. II. Ådahl M, Random iteration of isometries controlled by a Markov chain. Manuscript. III. Ådahl M, Melbourne I, Nicol M, Random iteration of Euclidean isometries. Nonlinearity 16 (2003) 977-987. IV. Johansson A, Ådahl M, Recurrence of a perturbed random walk and an iterated function system depending on a parameter. Manuscript. In the first paper we consider an iterated function system consisting of isometries on an unbounded metric space. Under suitable conditions it is proved that the random orbit {Zn} ∞n=0, of the iterations corresponding to an initial point Z0, “escapes to infinity" in the sense that P(Zn Є K) → 0, as n → ∞ for every bounded set K. As an application we prove the corresponding result in the Euclidean and hyperbolic spaces under the condition that the isometries do not have a common fixed point. In the second paper we let a Markov chain control the random orbit of an iterated function system of isometries on an unbounded metric space. We prove under necessary conditions that the random orbit \escapes to infinity" and we also give a simple geometric description of these conditions in the Euclidean and hyperbolic spaces. The results generalises the results of Paper I. In the third paper we consider the statistical behaviour of the reversed random orbit corresponding to an iterated function system consisting of a finite number of Euclidean isometries of Rn. We give a new proof of the central limit theorem and weak invariance principles, and we obtain the law of the iterated logarithm. Our results generalise immediately to Markov chains. Our proofs are based on dynamical systems theory rather than a purely probabilistic approach. In the fourth paper we obtain a suficient condition for the recurrence of a perturbed (one-sided) random walk on the real line. We apply this result to the study of an iterated function system depending on a parameter and defined on the open unit disk in the complex plane.

We compare three dependence coefficients expressed in terms of conditional expectations, and we study their behaviour in various situations. Next, we give a new covariance inequality involving the weakest of those coefficients, and we compare this bound to that obtained by Rio (Ann. Inst. H. Poincare Probab. Statist. 29 (1993) 587–597) in the strongly mixing case. This new inequality is used to derive sharp limit theorems, such as Donsker's invariance principle and Marcinkiewicz's strong law. As a consequence of a Burkholder-type inequality, we obtain a deviation inequality for partial sums.

We prove a central limit theorem for strongly mixing sequences under a sharp sufficient condition which combines the rate of the strong mixing coefficient with the quantile function. The result improves on all earlier central limit theorems for this type of dependence and answers a conjecture raised by Bradley in 1997. ¶ Moreover, we derive the corresponding functional central limit theorem.

The main purpose of this paper is to draw attention to a simple and useful general "continuity theorem" type result, from which a great deal of limit theorems follow as almost immediate consequences. As an example, we give a new very short and transparent proof of the recent result by H. Teicher and C. Hagwood (A multidimensional CLT for maxima of normed sums); in fact, a much more general assertion is proved here. Another application of the main result establishes a correspondence between the convergence of empirical and quantile processes. (A similar result holds for the renewal and partial sums processes.)

We count the number of times a random walk exits from a square root boundary and show that the normalized counting process and the normalized random walk converge jointly in law to a "local time," whose inverse is a stable subordinator of a known index, and a Brownian motion. The study of this limit process leads to some precise sample path properties of Brownian motion. These properties improve earlier results of Dvoretsky and Kahane on the existence of small oscillations in the Brownian path.