Paul Doukhan

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 representation are examples of the models considered. We establish Marcinkiewicz-Zygmund, Rosenthal and exponential inequalities for general sequences of centered random variables. Inequalities are stated in terms of the decay rate for the covariance of products of the initial random variables subject to the condition that the gap of time between both products tends to infinity. As applications of those notions, we obtain a version of the functional CLT and an invariance principle for the empirical process

A New Covariance Inequality and Applications

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 Donskers invariance principle and Marcinkiewiczs strong law. As a consequence of a Burkholder-type inequality, we obtain a deviation inequality for partial sums.

WEAK DEPENDENCE: MODELS AND APPLICATIONS TO ECONOMETRICS

In this paper we discuss weak dependence and mixing properties of some popular models. We also develop some of their econometric applications. Autoregressive models, autoregressive conditional heteroskedasticity (ARCH) models, and bilinear models are widely used in econometrics. More generally, stationary Markov modeling is often used. Bernoulli shifts also generate many useful stationary sequences, such as autoregressive moving average (ARMA) or ARCH(∞) processes. For Volterra processes, mixing properties obtain given additional regularity assumptions on the distribution of the innovations.We recall associated probability limit theorems and investigate the nonparametric estimation of those sequences.We first thank the editor for the huge amount of additional editorial work provided for this review paper. The efficiency of the numerous referees was especially useful. The error pointed out in Hall and Horowitz (1996) was the origin of the present paper, and we thank the referees for asking for a more detailed treatment of a correct proof for this paper in Section 2.3. Also we thank Marc Henry and Rafal Wojakowski for a very careful rereading of the paper. An anonymous referee has been particularly helpful in the process of revision of the paper. The authors thank him for his numerous suggestions of improvement, including important results on negatively associated sequences and a thorough update in standard English.

A triangular central limit theorem under a new weak dependence condition

We use a new weak dependence condition from Doukhan and Louhichi (Stoch. Process. Appl. 1999, 84, 313-342) to provide a central limit theorem for triangular arrays; this result applies for linear arrays (as in Peligrad and Utev, Ann. Probab. 1997, 25(1), 443-456) and standard kernel density estimates under weak dependence. This extends on strong mixing and includes non-mixing Markov processes and associated or Gaussian sequences. We use Lindeberg method in Rio (Probab. Theory Related Fields 1996, 104, 255-282).

A simple integer-valued bilinear time series model

In this paper, we extend the integer-valued model class to give a nonnegative integer-valued bilinear process, denoted by INBL(p,q,m,n), similar to the real-valued bilinear model. We demonstrate the existence of this strictly stationary process and give an existence condition for it. The estimation problem is discussed in the context of a particular simple case. The method of moments is applied and the asymptotic joint distribution of the estimators is given: it turns out to be a normal distribution. We present numerical examples and applications of the model to real time series data on meningitis and Escherichia coli infections.

Functional Estimation of a Density Under a New Weak Dependence Condition

The purpose of this paper is to prove, through the analysis of the behaviour of a standard kernel density estimator, that the notion of weak dependence defined in a previous paper (cf. Doukhan & Louhichi, 1999) has sufficiently sharp properties to be used in various situations. More precisely we investigate the asymptotics of high order losses, asymptotic distributions and uniform almost sure behaviour of kernel density estimates. We prove that they are the same as for independent samples (with some restrictions for a.s. behaviours). Recall finally that this weak dependence condition extends on the previously defined ones such as mixing, association and it allows considerations of new classes such as weak shifts processes based on independent sequences as well as some non‐mixing Markov processes.

Functional central limit theorem for the empirical process of short memory linear processes

We prove the functional central limit theorem for the empirical distribution function of a stationary causal moving average sequence with absolutely summable weights.

Asymptotics of weighted empirical processes of linear fields with long-range dependence

We discuss the asymptotic behavior of weighted empirical processes of stationary linear random fields in Z d with long-range dependence. It is shown that an appropriately standardized empirical process converges weakly in the uniform-topology to a degenerated process of the form fZ ,w hereZ is a standard normal random variable and f is the marginal probability density of the underlying random field.  2002 Editions scientifiques et medicales Elsevier SAS MSC: 60F17; 60G60 RESUME. - Nous etudions le comportement asymptotique du processus empirique pondere pour un champ lineaire stationnaire a longue dependance sur Z d . Nous montrons que ce processus convenablement normalise converge faiblement pour la topologie uniforme vers un processus degenere de la forme fZ ,o uZ est une variable normale standard et f est la densite de la marginale du champ considere.  2002 Editions scientifiques et medicales Elsevier SAS

Rates in the Empirical Central Limit Theorem for Stationary Weakly Dependent Random Fields

A weak dependence condition is derived as the natural generalization to random fields on notions developed in Doukhan and Louhichi (1999). Examples of such weakly dependent fields are defined. In the context of a weak dependence coefficient series with arithmetic or geometric decay, we give explicit bounds in Prohorov metric for the convergence in the empirical central limit theorem. For random fields indexed by &Zopfd, in the geometric decay case, rates have the form n−1/(8d+24)L(n), where L(n) is a power of log(n).

The notion of ψ -weak dependence and its applications to bootstrapping time series

We give an introduction to a notion of weak dependence which is more general than mixing and allows to treat for example processes driven by discrete innovations as they appear with time series bootstrap. As a typical example, we analyze autoregressive processes and their bootstrap analogues in detail and show how weak dependence can be easily derived from a contraction property of the process. Furthermore, we provide an overview of classes of processes possessing the property of weak dependence and describe important probabilistic results under such an assumption.