# Paul DoukhanCYU

Paul Doukhan

Professor

http://doukhan.perso.cyu.fr/ecodep.html

## About

194

Publications

28,637

Reads

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Introduction

Advisor for 20 PhD/Habilitation theses, I am involved into dependence structures. I introduced weak dependence with Sana Louhichi beyond mixing, to model real data.
It includes incomplete data, integer valued models and high dimensional data. Features of the real data are both their non stationarity, their dimension and the way they are sampled.
We also deal with applications to astronomy, insurance. Ecology is a main societal question whose approach necessitates the above techniques.

Additional affiliations

October 2002 - July 2008

September 2002 - March 2020

September 1993 - present

Education

July 1981 - May 1986

September 1979 - June 1981

October 1975 - September 2020

## Publications

Publications (194)

Let fn be the non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random vectors taking values in Rd. With some mild conditions, we establish sharp moderate deviations for the kernel density estimator. This means that we provide an equivalent for the tail probabilities of th...

Taylor's power law (TL) or fluctuation scaling has been verified empirically for the abundances of many species, human and non-human, and in many other fields including physics, meteorology, computer science, and finance. TL asserts that the variance is directly proportional to a power of the mean, exactly for population moments and, whether or not...

In this paper, we introduce a class of processes that contains many natural examples. The interesting feature of such type processes lays on its infinite memory that allows it to record a quite ancient history. Then, using the martingale decomposition method, we establish some deviation and moment inequalities for separately Lipschitz functions of...

Let (Zn)n≥0 be a supercritical Galton–Watson process. The Lotka–Nagaev estimator Zn+1/Zn is a common estimator for the offspring mean. In this paper, we establish some Cramér moderate deviation results for the Lotka–Nagaev estimator via a martingale method. Applications to construction of confidence intervals are also given.

The first motivation of this paper is to study stationarity and ergodic properties for a general class of time series models defined conditional on an exogenous covariates process. The dynamic of these models is given by an autoregressive latent process which forms a Markov chain in random environments. Contrarily to existing contributions in the f...

We propose a vector auto-regressive model with a low-rank constraint on the transition matrix. This model is well suited to predict high-dimensional series that are highly correlated, or that are driven by a small number of hidden factors. While our model has formal similarities with factor models, its structure is more a way to reduce the dimensio...

We consider integer-valued GARCH processes, where the count variable conditioned on past values of the count and state variables follows a so-called Skellam distribution. Using arguments for contractive Markov chains we prove that the process has a unique stationary regime. Furthermore, we show asymptotic regularity ($\beta$-mixing) with geometrica...

This paper aims at providing statistical guarantees for a kernel based estimation of time varying parameters driving the dynamic of local stationary processes. We extend the results of Dahlhaus et al. (2018) considering the local stationary version of the infinite memory processes of Doukhan and Wintenberger (2008). The estimators are computed as l...

We are studying linear and log-linear models for multivariate count time series data with Poisson marginals. For study- ing the properties of such processes we develop a novel conceptual framework which is based on copulas. Earlier contributions impose the copula on the joint distribution of the vector of counts by employing a continuous exten- sio...

This article proposes an optimal and robust methodology for model selection. The model of interest is a parsimonious alternative framework for modeling the stochastic dynamics of mortality improvement rates introduced by Doukhan et al. (2017). The approach models mortality improvements using a random field specification with a given causal structur...

Discrete time trawl processes constitute a large class of time series parameterized by a trawl sequence (a j) j$\in$N and defined though a sequence of independent and identically distributed (i.i.d.) copies of a continuous time process ($\gamma$(t)) t$\in$R called the seed process. They provide a general framework for modeling linear or non-linear...

The existing literature on extremal types theorems for stationary random processes and fields is, until now, developed under either mixing or “Coordinatewise (Cw)-mixing” conditions. However, these mixing conditions are very restrictives and difficult to verify in general for many models. Due to these limitations, we extend the existing theory, con...

In this paper, we adapt a data-driven smooth test to the comparison of the marginal distributions of two independent, short or long memory, strictly stationary linear sequences. Some illustrations are shown to evaluate the performances of our test.

In this paper, we adapt a data-driven smooth test to the comparison of the marginal distributions of two independent, short or long memory, strictly stationary linear sequences. Some illustrations are shown to evaluate the performances of our test.

We consider a class of non-homogeneous Markov chains, that contains many natural examples. Next, using martingale methods, we establish some deviation and moment inequalities for separately Lipschitz functions of such a chain, under moment conditions on some dominating random variables.

We consider a class of non-homogeneous Markov chains, that contains many natural examples. Next, using martingale methods, we establish some deviation and moment inequalities for separately Lipschitz functions of such a chain, under moment conditions on some dominating random variables.

Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on inequalities and learning theory for time series, in the past 15 years. However, for the non independent case, almo...

We propose a vector auto-regressive (VAR) model with a low-rank constraint on the transition matrix. This new model is well suited to predict high-dimensional series that are highly correlated, or that are driven by a small number of hidden factors. We study estimation, prediction, and rank selection for this model in a very general setting. Our me...

Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on inequalities and learning theory for time series, in the past 15 years. However, for the non independent case, almo...

We study nonlinear mixtures of integer-valued ARCH type models for count time series data. We investigate the theoretical properties of these processes and we prove ergodicity and stationarity, under minimal assumptions. The model can be generalized by including a GARCH component but we show that such inclusion can be accommodated by an ARCH model...

This is a research project in Valparaiso for modelling astronomical data through non stationary models

We introduce a class of discrete time stationary trawl processes taking real or integer values and written as sums of past values of independent ‘seed’ processes on shrinking intervals (‘trawl heights’). Related trawl processes in continuous time were studied in Barndorff-Nielsen et al. (2011, 2014). In the case when the trawl function decays as a...

This chapter describes a simple Gaussian limit theory; namely we restate simple central limit theorems together with applications and moment/exponential inequalities for partial sums behaving asymptotically as Gaussian random variables. A relevant reference for the whole chapter is Petrov (Limit theorems of probability theory. Sequences of independ...

We propose an overview of the notions of dependence in this chapter, good references are Doukhan et al. (Theory and applications of long-range dependence. Birkhaüser, Boston, 2002b) for long-range dependence, and Doukhan (Mixing: properties and examples. Lecture notes in statistics. Springer, New York, 1994) and Dedecker et al. (Weak dependence: wi...

The notion of association, or positive correlation, was naturally introduced in two different fields: reliability (Esary, Proschan, Walkup in Annal Math Stat 38:1466–1474, 1967) and statistical physics (Fortuin, Kasteleyn, Ginibre in Commun Math Phys 22–2:89–103, 1971) to model a tendency that the coordinates of a vector valued random variable admi...

We consider stationary sequences generated through independent identically distributed \((\xi _n)_{n\in \mathbb {Z}}\). A reference is Brockwell and Davis (Time series: theory and methods. Springer, New York, 1991). Such models are natural in signal theory since they appear through linear filtering of a white noise. The usual setting is that \((\xi...

This chapter aims at describing stationary sequences generated from independent identically distributed samples \((\xi _n)_{n\in \mathbb {Z}}\). Most of the material in this chapter is specific to this monograph so that we do not provide a global reference. However Rosenblatt (Stationary processes and random fields. Birkhäuser, Boston, 1985) perfor...

The present chapter deals with the standard notion of stochastic independence. This is a crucial concept, since this monograph aims to understand how to weaken it, in order to define asymptotic independence. We discuss in detail the limits of this idea through various examples and counter-examples.

This chapter is devoted to moment methods. The use of moments relies on their importance in deriving asymptotic of several estimators, based on moments and limit distributions. Cumulants are linked with spectral or multispectral estimation which are main tools of time series analysis. $$g(\lambda )=\sum _{k=-\infty }^\infty \mathrm {Cov}\,(X_0,X_k)...

Some bases for the theory of time series are given below. The chapter deals with the widely used assumption of stationarity which yields a simpler theory for time series. This concept is widely considered in Rosenblatt (Stationary processes and random fields, Birkhäuser, Boston, 1985) and in Brockwell and Davis (Time series: theory and methods, 2nd...

Gaussian distributions (Appendix A) are natural and play a special role in the field of probability theory since they appear as limit distributions from the CLT (Theorem 2.1.1, Lemma 11.5.1). Gaussian linear spaces admit a simple geometric property.

This chapter introduces some simple ideas. We investigate conditions on time series such that the standard limit theorems obtained for independent identically distributed sequences still hold. After a general introduction to weak-dependence conditions an example states the fact that the most classical strong-mixing condition from (Rosenblatt (1956)...

Long-range dependent (LRD) phenomena were first exhibited by Hurst for hydrology purposes. This phenomenon occurs from the superposition of independent sources, e.g. confluent rivers provide this behaviour (see Fig. 4.2). Such aggregation procedures provide this new phenomenon. Hurst (Trans Am Soc Civ Eng 116:770–799, 1951) originally determined th...

Many statistical procedures are derived from probabilistic inequalities and results; such procedures may need more precise bounds as this is proved in the present chapter for the independent case. Basic notations are those from Appendix B.1. Developments may be found in van der Vaart (Asymptotic statistics, Cambridge University Press, Cambridge, 19...

https://www.springer.com/fr/book/9783319769370
This book presents essential tools for modelling non-linear time series. The first part of the book describes the main standard tools of probability and statistics that directly apply to the time series context to obtain a wide range of modelling possibilities. Functional estimation and bootstrap are...

The existing literature on extremal types theorems for stationary random processes and fields are, until now, developed under either mixing or “Coordinate- wise (Cw)-mixing” conditions. However, these mixing conditions are very restric- tives and difficult to verify in general. In this context, we provide an extremal types theorem for stationary ra...

In this article we propose a quasi-Whittle estimator for parametric families of time series models in the presence of missing data. This estimator extends results to the incompletely observed case. This extension is valid to non-Gaussian and nonlinear models. It also allows us to bound the variance of an associated quasiperi-odogram. A simulation s...

We prove existence and uniqueness of a stationary distribution and absolute regularity for nonlinear GARCH and INGARCH models of order (p,q). In contrast to previous work we impose, besides a geometric drift condition, only a semi-contractive condition which allows us to include models which would be ruled out by a fully contractive condition. This...

This article proposes a parsimonious alternative approach for modeling the stochastic dynamics of mortality rates. Instead of the commonly used factor-based decomposition framework, we consider modeling mortality improvements using a random field specification with a given causal structure. Such a class of models introduces dependencies among adjac...

Extending the ideas of [7], this paper aims at providing a kernel based non-parametric estimation of a new class of time varying AR(1) processes (Xt), with local stationarity and periodic features (with a known period T), inducing the definition Xt = at(t/nT)X t--1 + $\xi$t for t $\in$ N and with a t+T $\not\equiv$ at. Central limit theorems are es...

We are studying the problems of modeling and inference for multivariate count time series data with Poisson marginals. The focus is on linear and log-linear models. For studying the properties of such processes we develop a novel conceptual framework which is based on copulas. However, our approach does not impose the copula on a vector of counts;...

With the simple example of an asymmetric ARCH(1) model as a pretext, we introduce some of the main tools for weak dependence conditions introduced in [7]. The power of weak dependence is shown up for this very elementary model. This a special case of the infinite memory models in [8]. Asymptotic properties of a moment based parametric estimation do...

Gene promoters have variable repartition of AGCT nucleotides according to some probabilistic behaviours essentially depending on their position in a string. The paper aims to provide a model for this configuration. With this model we derive non-uniform confidence bounds for those probability distributions in the strings. A uniform bound deriving fr...

We introduce a class of discrete time stationary trawl processes taking real or integer values and written as sums of past values of independent `seed' processes on shrinking intervals (`trawl heights'). Related trawl processes in continuous time were studied in Barndorff-Nielsen (2011) and Barndorff-Nielsen et al. (2014), however in our case, the...

We propose a general definition for weak dependence of point processes as an alternative to mixing definitions. We give examples of such weak dependent point processes for the families of Neyman Scott processes or Cox processes. For these processes, we consider the empirical estimator of the empty space function . Using the general setting of the w...

We consider here together the inference questions and the change-point problem in a large class of Poisson autoregressive models (see Tjøstheim, 2012 [34]). The conditional mean (or intensity) of the process is involved as a non-linear function of it past values and the past observations. Under Lipschitz-type conditions, it can be written as a func...

The notion of a phantom distribution function (phdf) was introduced by
O'Brien (1987). We show that the existence of a phdf is a quite common
phenomenon for stationary weakly dependent sequences. It is proved that any
$\alpha$-mixing stationary sequence with continuous marginals admits a
continuous phdf. Sufficient conditions are given for stationa...

Understanding the way extreme values do cluster in the case of time series is an essential problem for extreme values theory and risk management. Drees and Rootzen (2010) provided a deep solution to this problem.
Anyway the existing literature on functional central limit theorems (FCLT) for empirical processes of cluster functionals (EPCF) is, unti...

In this paper, we propose a model-free bootstrap method for the empirical process under absolute regularity. More precisely, consistency of an adapted version of the so-called dependent wild bootstrap, which was introduced by Shao (2010) and is very easy to implement, is proved under minimal conditions on the tuning parameter of the procedure. We s...

We study the almost sure limiting behavior of record times and the number of records, respectively, in a (so-called) \(F^\alpha \)-scheme. It turns out that there are certain “dualities” between the latter results, that is, under rather general conditions strong laws for record times can be derived from the corresponding ones for the number of reco...

In this paper we propose a smooth test of comparison for the marginal distributions of strictly stationary dependent bivariate sequences. We first state a general test procedure and several cases of dependence are then investigated. The test is applied to both simulated data and real datasets.

We discuss a class of conditionally heteroscedastic time series models
satisfying the equation $r_t= \zeta_t \sigma_t$, where $\zeta_t$ are
standardized i.i.d. r.v.'s and the conditional standard deviation $\sigma_t$ is
a nonlinear function $Q$ of inhomogeneous linear combination of past values
$r_s, s<t$ with coefficients $b_j$. The existence of s...

We consider generalized linear models for regression modeling of count time series.
We give easily verifiable conditions for obtaining weak dependence for such models.
These results enable the development of maximum likelihood inference under minimal
conditions. Some examples which are useful to applications are discussed in detail.

We determine the almost sure and central limiting behaviour of the number of records and record times for the F
α
-scheme. Elementary methods are used to obtain general results. The basic results are extended to a random environments model.

We consider here together the inference questions and the change-point
problem in Poisson autoregressions (see Tj{\o}stheim, 2012). The conditional
mean (or intensity) of the process is involved as a non-linear function of it
past values and the past observations. Under Lipschitz-type conditions, it is
shown that the conditional mean can be written...

We investigate the relationship between weak dependence and mixing for discrete valued processes. We show that weak dependence implies mixing conditions under natural assumptions. The results specialize to the case of Markov processes. Several examples of integer valued processes are discussed and their weak dependence properties are investigated b...

This paper deals with a general class of observation-driven time series
models with a special focus on time series of counts. We provide conditions
under which there exist strict-sense stationary and ergodic versions of such
processes. The consistency of the maximum likelihood estimators is then derived
for well- specified and misspecified models.

We consider generalized linear models for regression modeling of count time series. We give easily verifiable conditions for obtaining weak dependence for such models. These results enable the development of maximum likelihood inference under minimal conditions. Some examples which are useful to applications are discussed in detail.

We give rates of convergence in the strong invariance principle for
stationary sequences satisfying some projective criteria. The conditions are
expressed in terms of conditional expectations of partial sums of the initial
sequence. Our results apply to a large variety of examples, including mixing
processes of different kinds. We present some appl...

The aim of this paper is to provide a comprehensive introduction for the study of ℓ1-penalized estimators in the context of dependent observations. We define a general ℓ1-penalized estimator for solving problems of stochastic optimization. This estimator turns out to be the LASSO [Tib96] in the regression estimation setting. Powerful theoretical gu...

The aim of this paper is to provide a comprehensive introduction for the
study of L1-penalized estimators in the context of dependent
observations. We define a general $\ell_{1}$-penalized estimator for
solving problems of stochastic optimization. This estimator turns out to
be the LASSO in the regression estimation setting. Powerful theoretical
gu...

Fazekas and Klesov (2000) found conditions for almost sure con-vergence rates in the law of large numbers that effectively can be applied if maximal inequalities are available. In the spirit of Móricz (1976), we aim at using those conditions in a weakly dependent framework, and this trick is proved to be quite efficient, first in the standard law o...

This paper provides extensions of the work on subsampling by Bertail et al. (2004) for strongly mixing case to weakly dependent case by application of the results of Doukhan and Louhichi (1999). We investigate properties of smooth and rough subsampling estimators for distributions of converging and extreme statistics when the underlying time series...

This paper provides extensions of the work on subsampling by Bertail et al.in J. Econ. 120:295–326 (2004) for strongly mixing case to weakly dependent case by application of the results of Doukhan and Louhichi in Stoch. Proc.
Appl. 84:313–342 (1999). We investigate properties of smooth and rough subsampling estimators for sampling distributions of...

Self-normalized central limit theorems are important for statistical pur-poses. A simple way to achieve them is to consider estimations of the limit variance; this expression writes as a complicated covariance series under weak dependence. Using an argument of Carlstein (1986), we work out this program for a new procedure, in the case of vector val...

Fazekas and Klesov (2000) found conditions for almost sure convergence rates in the law of large numbers that effectively can be applied if maximal inequalities are available. In the spirit of M´oricz (1976), we aim at using those conditions in a weakly dependent framework, and this trick is proved to be quite efficient, first in the standard law o...

This paper provides extensions of the work on subsampling by Bertail et al. (2004) for strongly mixing case to weakly dependent case by application of the results of Doukhan and Louhichi (1999). We investigate properties of smooth and rough subsampling estimators for distributions of converging and extreme statistics when the underlying time series...

This volume collects recent works on weakly dependent, long-memory and multifractal processes and introduces new dependence measures for studying complex stochastic systems. Other topics include the statistical theory for bootstrap and permutation statistics for infinite variance processes, the dependence structure of max-stable processes, and the...

We state a multidimensional Functional Central Limit Theorem for weakly dependent random vectors. We apply this result to
copulas. We get the weak convergence of the empirical copula process and of its smoothed version. The finite dimensional convergence
of smoothed copula densities is also proved. A new definition and the theoretical analysis of c...

We prove the existence of a weakly dependent strictly stationary solution of the equation Xt=F(Xt−1,Xt−2,Xt−3,…;ξt) called a chain with infinite memory. Here the innovationsξt constitute an independent and identically distributed sequence of random variables. The function F takes values in some Banach space and satisfies a Lipschitz-type condition....

We prove uniform convergence results for the integrated periodogram of a weakly dependent time series, namely a strong law of large numbers and a central limit theorem. These results are applied to Whittle's parametric estimation. Under general weak-dependence assumptions, the strong consistency and asymptotic normality of Whittle's estimate are es...

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 deri...

Ratios of random variables often appear in probability and statistical applications. We aim to approximate the moments of such ratios under several dependence assumptions. Extending the ideas in Collomb [C. R. Acad. Sci. Paris 285 (1977) 289--292], we propose sharper bounds for the moments of randomly weighted sums and for the $L^p$-deviations from...

The paper is devoted to recall weak dependence conditions from Dedecker et al. (Weak dependence, examples and applications.
Lecture Notes in Statistics, vol 190, 2007)’s monograph; the main basic results are recalled here and we go further in some
new applications. We develop here several models of weakly dependent processes and random fields. Amon...

We prove a general functional central limit theorem for weak dependent time series. A very large variety of models, for instance, causal or non causal linear, ARCH($\infty$), bilinear, Volterra processes, satisfies this theorem. Moreover, it provides numerous application as well for bounding the distance between the empirical mean and the Gaussian...

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 such as stable Markov chains, dynamical systems or more complicated models, nonlinear, non-Markovian, and heteroskedastic models with infinite memory. Mos...

Doukhan and Louhichi [P. Doukhan, S. Louhichi, A new weak dependence condition and application to moment inequalities, Stochastic Process. Appl. 84 (1999) 313-342] introduced a new concept of weak dependence which is more general than mixing. Such conditions are particularly well suited for deriving estimates for the cumulants of sums of random var...

In this paper, a very useful lemma (in two versions) is proved: it
simplifies notably the essential step to establish a Lindeberg
central limit theorem for dependent processes. Then, applying this
lemma to weakly dependent processes introduced in Doukhan and
Louhichi (1999), a new central limit theorem is obtained for
sample mean or kernel density...

Giraitis and Surgailis [Giraitis, L., Surgailis, D., 2002, ARCH-type bilinear models with double long memory. Stochastic Processes and their Applicatons, 100, 275–300.] introduced ARCH-type bilinear models for their specific long-range dependence properties. We rather consider weak-dependence properties of these models. The computation of mixing co...

Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally integrated filters A(d), B(d) depending on an arbitrary sequence d = (d
t
)
t∈ℤ of real numbers; if the parameter sequence is constant d
t
≡ d, then both filters A(d) and B(d) reduce to the usual fractional integration operator (1 − L)−d
. They also st...

## Questions

Questions (4)

I can no more send messages on researchgate

when I click on send, the interface tells that message could not be sent

How to test the randomness and the equidistribution for many series of sequences of integers?

The first name of my coauthor in

Weak dependence: an introduction through asymmetric arch models

was changed

Thus authors are not the right ones

non stationarity is sometimes considered in the sense of econometricians as unit root problems 5PCB Philipps et al.).

A question would be to unify "reasonable" notions of non stationarity, eg local stationarity (eg Dahlhaus and followers), periodic or seasonal (eg Dickey, Mohamedou Ould Haye, Viano, Leskow), unit roots.

For periodic behaviors an additional attractive question is to fit periods (even if the continuous time case is even more relevant and a problem is then to fit periods

Many other non stationarity notions, such as random walks in random environment (eg Snitzmann) are extremely attractive: Anyway they dont necessarily features of real data

The idea of the present project is more to develop (non parametric or parametric) models featuring the main properties of real data sets to be fitted

Certainly the validity of techniques should be provided from real data analysis as this was emblematically done in a contradictory paper by Mikosch and Starica opposing long range dependent models to models with a linear trend