Sana Louhichi

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

Weak convergence for empirical processes of associated sequences

Abstract We establish a weak convergence theorem for empirical processes of stationary and associated random variables having the uniform marginal distribution. To carry out the proof, we develop a tightness criterion for the empirical process constructed from any stationary sequence fulfilling a suitable moment inequality. We apply the result to stationary non mixing moving average sequences with positive coefficients. Based on this class of linear processes, we compare mixing and association.

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.

Maximal Inequalities and Empirical Central Limit Theorems

This work presents some recent developments concerning empirical central limit theorems for dependent sequences. We try to give general conditions under which an adaptive truncation in the chaining procedure works. We show that these conditions are satisfied for a large class of mixing processes satisfying suitable moment inequalities.

Marcinkiewicz–Zygmund Strong Laws for Infinite Variance Time Series

We study rate of convergence in the strong law of large numbers for finite and infinite variance time series in both contexts of weak and strong dependence.

Exponential Growth of Bifurcating Processes with Ancestral Dependence

Branching processes are classical growth models in cell kinetics. In their construction, it is usually assumed that cell lifetimes are independent random variables, which has been proved false in experiments. Models of dependent lifetimes are considered here, in particular bifurcating Markov chains. Under the hypotheses of stationarity and multiplicative ergodicity, the corresponding branching process is proved to have the same type of asymptotics as its classic counterpart in the independent and identically distributed supercritical case: the cell population grows exponentially, the growth rate being related to the exponent of multiplicative ergodicity, in a similar way as to the Laplace transform of lifetimes in the i.i.d. case. An identifiable model for which the multiplicative ergodicity coefficients and the growth rate can be explicitly computed is proposed.

Estimation de la densité d'une suite faiblement dépendante

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. [4]) has sharp properties enough to be used in various situations. This weak dependence condition extends the previously defined ones such as mixing, association and it allows to consider new classes such as weak shift processes based on independent sequences as well as some non mixing Markov processes.

Rosenthal's inequality for LPQD sequences

We note that Rosenthals type inequalities for LPQD random variables (r.v.s) are written along the line of Birkel [Birkel, T., 1988. Moment bounds for associated sequences. Ann. Probab. 16, 1184-1193]. A generalization of this result is obtained.

A cluster-limit theorem for infinitely divisible point processes

In this article, we consider a sequence (N n ) n≥1 of point processes, whose points lie in a subset E of ℝ \{0} and satisfy an asymptotic independence condition. Our main result gives some necessary and sufficient conditions for the convergence in the distribution of (N n ) n≥1 to an infinitely divisible point process N. As applications, we discuss the exceedance processes and point processes based on regularly varying sequences.

On the asymptotic normality of the R-estimators of the slope parameters of simple linear regression models with associated errors

ABSTRACT The purpose of this paper is to prove, under mild conditions, the asymptotic normality of the rank estimator of the slope parameter of a simple linear regression model with stationary associated errors. This result follows from a uniform linearity property for linear rank statistics that we establish under general conditions on the dependence of the errors. We prove also a tightness criterion for weighted empirical process constructed from associated triangular arrays. This criterion is needed for the proofs which are based on that of Koul [Behavior of robust estimators in the regression model with dependent errors. Ann Stat. 1977;5(4):681–699] and of Louhichi [Louhichi S. Weak convergence for empirical processes of associated sequences. Ann Inst Henri Poincaré Probabilités Statist. 2000;36(5):547–567].