Program

CONFERENCE

Paul DOUKHAN

 

 

"Dependence, Limit Theorems and Applications"

 

11, 12 and 13 May 2015

 

PREVISIONAL SCHEDULE

 

 

Monday 11 May 2015

 

            10h00-10h20: Opening

            10h20-11h00: D. Surgailis (Academy of Sciences of Lithuania)

Title: Scaling transition for long-range dependent random fields

Abstract: \cite{ps2014}, \cite{ps2015} introduced the notions of scaling  transition and distributional long-range dependence for stationary random fields  $X$ on ${\mathbb{Z}}^2$ whose normalized partial sums on rectangles with sides growing at rates $O(n) $ and $O(n^{\gamma})$ tend to an operator scaling random field $V_\gamma $ on ${\mathbb{R}}^2$, for any $\gamma >0$. The scaling  transition is characterized by the fact that there exists a unique $\gamma_0 >0$ such that the scaling limits
$V_\gamma$ are different and do not depend on $\gamma $ for $\gamma > \gamma_0$ and $\gamma < \gamma_0$.  It appears that scaling transition is a general phenomenon, suggesting an exciting new area in spatial research. The talk presents some new results about the existence and characterization of scaling  transition for isotropic and anisotropic linear models and some other classes of long-range dependent random fields.   

            11h00-11h20: Coffee Break 

            11h20-12h00: P. Massart (University Paris XI and IUF)  

Title: Estimator selection: methods and calibration

Abstract: There is a huge literature devoted to the topic of estimator selection, both from a theoretical and a practical view point. Estimator selection  methods  typically involve tuning parameters which drastically influence the behavior of the methods. Therefore the choice of these tuning parameters is an important issue. It is often the case that performance bounds for estimator selection methods are over pessimistic or not precise enough to provide a valuable guide-line for practitioners. Fortunately computer intensive simulations can be used to understand how to calibrate these tuning parameters in practice but nevertheless from a mathematical view point, it is desirable to better understand the issue of calibrating selection methods. We shall see that in a number of situations that include penalized model selection, it is possible to reduce the gap between theory and practice by providing data-driven rules for choosing tuning parameters which are based on sharp upper and lower performance bounds.

            12h00-12h40: I. Nourdin (University of Luxembourg)

Title: Gaussian Phase Transitions for Conic Intrinsic Volumes


Abstract: Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, for instance in the compressed sensing theory. In this talk I will explain why, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution and lead to a phase transition. This talk is based on a joint work with Larry Goldstein (Southern California) and Giovanni Peccati (Luxembourg).


                Lunch in I.H.P.


            14h00-14h40: Q. Liu (University of Vannes)   

Title: Sharp large deviation inequalities and expansions for sums of independent random variables


Abstract: We show sharp bounds for probabilities of large deviations for sums of independent random variables satisfying Bernstein’s condition.  One such bound is very close to the tail of the standard Gaussian law in certain case; other bounds improve Bennett and Hoeffding’s inequalities by adding missing factors in the spirit of Talagrand (1995). We also complete Talagrand’s inequality by giving a lower bound of the same form.  As a corollary, we obtain large deviation expansions similar to those of Cramér,  Bahadur-Rao and Sakhanenko.
The talk is based on a joint work with Xiequan Fan and Ion Grama.

            14h40-15h20: E. Moulines (Telecom ParisTech)

Title: Subgeometric rates of convergence in Wasserstein distance for Markov  chains


Abstract: We provide sufficient conditions for the existence of the  invariant distribution and for subgeometric rates of convergence in Wasserstein distance for general state-space Markov chains which are  (possibly) not irreducible.  Compared to (Butkovsky, 2013, AoAP) our approach is based on a coupling construction  which allows to retrieve (up to constants) rates of convergence matching those   previously reported for convergence in total variation in (Douc, M.,  Soulier, 2007). Our results are applied to establish the subgeometric ergodicity in Wasserstein distance of non-linear autoregressive models and also of an infinite dimensional MCMC algorithm, used to solve an elliptic inverse problem in Sobolev space (the   pre-conditioned Crank-Nicolson Markov chain Monte Carlo algorithm, introduced by Stuart).

            15h20-16h00: I. Grama (University of Vannes)

Title: Estimation of extreme probabilities and quantiles from functional data

Abstract: Let $(F_t(x))_{t \in [0,T_{\max}],}$ be a family of distribution functions indexed in the  interval $[0,T_{\max}],$
where it is assumed that $F_t$ are in the domain of attraction of the Fr\'echet law. We are interested in recovering the tail of $F_t(x)$ given the time $t \in [0,T_{\max}]$ from independent random variables $X_{t_1},\ldots,X_{t_n}$ with distributions $F_{t_1},\ldots,F_{t_n},$ For each $t\in [0,T_{\max}]$, we propose a nonparametric adaptive estimator for excess distribution function of $F_t$ over the threshold $\tau$ using a nonparametric kernel estimator of bandwidth $h$ based on the observations larger than $\tau$ in the interval $[t-h,t+h].$ The main challenge in this setting is to choose the threshold $\tau$ and the bandwidth $h$. We propose a new pointwise data driven procedure to choose the threshold $\tau$ and two selection procedures of the bandwidth $h$, one global based on a cross-validation approach and second one local based on a pointwise adaptive approach. Under regularity assumptions, we prove that the proposed non adaptive and adaptive estimators are consistent and we determine their rate of convergence. We study the proposed procedures by simulations and we give applications to an environmental data set.

            16h00-16h20: Coffee Break

            16h20-17h00: D. Tjosheim (University of Bergen)          

Title: Nonstationary processes with a threshold

Abstract: A review is given on Markov chain recurrence theory in the context of time series models that are both nonlinear and nonstationary. This is applied to threshold unit root processes and to a new type of threshold cointegration models.

            17h00-17h40: S. Louhichi (University of Grenoble 1)     

Title: Exponential growth of bifurcating processes with ancestral dependence

Abstract: 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 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 i.i.d.  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.

Joint work with Bernard Ycart

            17h40-18h20: J. Yao (University of Hong-Kong)

Title: Outliers in the spectrum of large sample covariance matrices


Abstract: Random matrix theory essentially is an attempt to understand the behaviour of the eigenvalues of a large random matrix. Because their number is large, say a few thousands, it is useless or impossible to examine these eigenvalues individually. Rather, interesting questions concern either their joint empirical distribution or a few particular among them such as the extreme eigenvalues (the largest or the smallest ones). While the study of the joint empirical distribution of large random matrices has been long standing and well developed since E.P. Wigner's work in the fifties of last century, the advances on the extreme eigenvalues have been much more recent. In this talk, I will concentrate on a specific class of spiked covariance matrices (finite-rank perturbed). These matrices have attracted much attention in the last fifteen years because of their "multidisciplinary" nature: indeed, related questions can be deeply theoretical or completely applied to real-life data analysis. For these matrices, we study the effect produced on the extreme eigenvalues by a finite-rank peerturbation of the popualtion covariance matrix. I will select some of these questions, mostly from a probabilistic and analytic perspective, but also from a data analysis perspective with some concrete application.

 

            Welcome Cocktail (IHP)

 

Tuesday 12 May 2015    

 

            09h00-09h40: J. Dedecker (University Paris 5)  

Title: Subgaussian concentration inequalities for geometrically ergodic Markov chains.


Abstract: We show that an irreducible aperiodic Markov chain is geometrically ergodic if and only if any separately bounded functional of the stationary chain satisfies an appropriate subgaussian deviation inequality from its mean.
Joint work with Sébastien Gouëzel (IRMAR, Université de Rennes 1).

            09h40-10h20: T. Mikosch (University of Copenhagen)

Title: Some results related to the solution to  equation in law $X=AX+B$


Abstract: The equation in law $X=AX+B$ for independent $X$ and $(A,B)$ has attracted a lot of attention over several decades. The article by Kesten (Act. Math. 1973) initiated a long series of research papers devoted to this topic. Given that the recurrence equation $X_t=A_tX_{t-1}+B_t$, $t\ge 0$, for an iid sequence $(A_t,B_t)$, $t\ge 0$, has a stationary solution $(X_t)$, the marginal distribution satisfies the equation in law $X=AX+B$. The time series $(X_t)$ has a lot of interesting properties. For example, under mild assumptions on $(A,B)$, the finite-dimensional distributions of $(X_t)$ are regularly varying (have power law tails). In turn, this property implies the infinite variance stable central limit theoreom, convergence of the point processes of the $X_t$'s, convergence of the  extremes, heavy-tailed large deviation results, etc. In this talk some of the more recent asymptotic results for $(X_t)$ will be presented.

            10h20-11h00: R. Dahlhaus (University of Heidelberg)  

Title: Volatility Decomposition and Online Volatility-Estimation with Nonlinear Market Microstructure Noise Models

Abstract: A technique for online estimation of spot volatility for high-frequency data is developed. The method uses a price model with time shift in combination with a nonlinear market microstructure noise model. A benefit of the model is that it leads to an identifiable decomposition of spot volatility into spot volatility per transaction and the trading intensity - thus highlighting the influence of trading intensity on volatility. The online algorithm uses a computationally efficient particle filter. It works directly on the transaction data and updates the volatility estimate immediately after the occurrence of a new transaction. It also allows for the approximation of the unknown efficient prices. For volatility estimation a nonparametric recursive EM algorithm is used. We neither assume that the transaction times are equidistant nor do we use interpolated prices. For the theoretical investigations of the estimates we present a theoretical framework with infill asymptotics.

Joint work with Jan. C. Neddermeyer and Sophon Tunyavetchakit

            11h00-11h20: Coffee Break 

            11h20-12h00: P. Alquier (ENSAE) 

Title: Bayesian Matrix Completion

Abstract: Low-rank matrix estimation from incomplete measurements recently received increased attention due to the emergence of several challenging applications (among the most famous, the Netflix challenge). Most known methods rely on penalized risk minimization. For algorithmic reasons, the penalty is usually related to the nuclear norm of the matrix. An as yet unexplored avenue of research is to develop Bayesian methodology for this type of problem. In a first time, I will introduce different priors on low-rank matrices. I will then discuss the consistency, the minimax-optimality and the practical relevance of the various Bayesian estimators.

            12h00-12h40: H. Dehling (University of Bochum)

Title: Empirical Process CLTs for Dependent Data


Abstract: In our talk we will first give a survey of results on empirical process central limit theorems for dependent data, starting from the early beginnings in the 1960s. We will then present a new technique for establishing empirical process CLTs, recently developed by Dehling, Durieu, and Volny, with applications to dynamical systems and Markov chains.


                Lunch in I.H.P.

 

             14h00-14h40: B. Delyon (University of Rennes 1)

Title: Acceleration of empirical means


Abstract:We discuss the following limit-theorem: Let (X1 , . . . , Xn ) be an i.i.d. sequence of random variables in R^d , and ϕ : R^d → R, then under reasonable regularity and integrability conditions
n^1/2 ( 1/n \sum_ i=1^n \phi(X_i)/\hat f^(i)(X_i) - \int \phi(x)dx ) \to 0 where \hat f^(i) is the leave-one-out kernel estimator of the density f of X_1.
It is a collaborative work with François Portier

             14h40-15h20: F. Gamboa (University of Toulouse)      

Title: Korovkin theorem and super resolution old and new results


Abstract: Super resolution consists in the reconstruction of sparse measures with partial information on a finite number of its generalized moments. In this talk, we will discuss both on the 90'ths results on super-resolution obtained by Paul and on recent works on this subject.

             15h20-16h00: C. Prieur (University of Grenoble 1)       

Title: Estimation for hypoelliptic diffusions.


Abstract:In this work, we are interested in harmonic oscillators perturbed with a gaussian white noise. More precisely, we consider Zt := (xt , yt ) ∈ R2d , t ≥ 0 governed by the following Ito stochastic differential equation:
$dxt = yt dt,   dyt = σ I dWt − (c(xt , yt )yt +\nabla V (xt ))dt$
We assume that the process is ergodic with a unique invariant probability measure µ, and that the convergence in the ergodic theorem is quick enough. We also discuss sufficient conditions for this. For such oscillators, we aim at studying inference issues such as the estimation of the density of the invariant probability measure µ, as far as the estimation of the drift or the variance term. One major issue in our study is that we work with incomplete data, observing only the first coordinate X. Thus we approximate the Y component by finite differences. Even in case the potential is the Duffing’s one V (x) = x4 /4 − x2 /2 (Kramers
oscillator) this problem is not easy. We focus on non-parametric inference, see [1,2,3,4].

             16h00-16h20: Coffee Break

             16h20-17h00: J.L. Prigent (University of Cergy-Pontoise)  

Title: On the Constant Proportion Portfolio Insurance (CPPI) Method


Abstract:First we briefly recall the portfolio insurance principles, the CPPI framework and the main properties of the unconditional multiples. Second, we present several models for the conditional multiple whose aim is to adapt the current exposition to market conditions by measuring the gap risk and following a traditional risk management philosophy.  Finally, we estimate these conditional multiples using parametric, semi-parametric and non-parametric methods and compare these different approaches.

             17h00-17h40: M. Neumann (University of Jena)  

Title: A model specification test for GARCH(1,1) processes'


Abstract: A consistent specification test for GARCH(1,1) models based on a test statistic of Cram\'er-von Mises type is proposed.
Since the limit distribution of the test statistic under the null hypothesis depends on unknown quantities in a complicated manner,
we use a model-based (semiparametric) bootstrap method to approximate critical values of the test.
Asymptotic correctness is shown by coupling arguments.

             17h40-18h20: J. Leon (University of Caracas)

Title: CLT for level functionals and applications.


Abstract:In this talk we will establish  a general method for obtaining a CLT for level functionals of stationary Gaussian process or fields. The procedure can be sketched in the following form.
a/ The use of the Rice's formula both for the expectation and the second factorial moment (or second moment) of the functional allows getting conditions under which such functionals are in $\mathbb L^2(\Omega)$.
b/ The approximation of the functional by means of a well defined occupation functional, permits to get an Itô-Wiener's expansion with precise coefficients.
c/ Under some $\mathbb L^1(dx)\cap\mathbb L^2(dx)$ hypothesis for the covariance  functions of the process and a careful use of the inequality of Miguel Arcones we get that the asymptotic variance of the normalized functional is bounded.
d/ This last result is used afterwards to show that for obtaining the CLT it is enough to work with a finite dimensional expansion into the chaos. Then an application of the Breuer-Major's Theorem in the modern version of Nourdin-Peccati-Podolskij gives the asymptotic normality.
e/ The talk contains two applications: firstly to the number of crossings of a stationary Gaussian process and  secondly to the Euler's characteristic of an excursion set of an isotropic Gaussian random field.

             19h30: Dinner in Restaurant "Le Tournebièvre", 65 Quai de la Tournelle, 75005 Paris

 

Wednesday 13 May 2015  

 

             09h00-09h40: A. Jakubowski (University of Torun)

Title: Weak dependence and the extremal zero stationary processes


Abstract: We apply the framework of weak dependence to asymptotic theory of maximal terms of stationary sequences. In particular we provide several models built on heavy-tailed elements, which have the extremal index zero, while they admit a continuous phantom distribution function in the sense of O'Brien. The models include the Lindley process, the random walk Metropolis algorithm and  an example which is non-ergodic.
This is a joint work with Paul Doukhan and Gabriel Lang.

             09h40-10h20: B. Bercu (University of Bordeaux)

Title: Concentration inequalities for martingales with statistical applications


Abstract: This talk is devoted to an overview of concentration inequalities for martingales. We will start with classical exponential inequalities for sums of independent random variables such as Hoeffding, Bennett and Bernstein inequalities. Then, we will focus our attention on exponential inequalities for martingales such as Azuma-Hoeffding, Freedman and De la Pena inequalities. We will also investigate more recent concentration inequalities for martingales and self-normalized martingales. Finally, we will finish the talk by some statistical applications on autoregressive processes, branching processes and random permutations.

             10h20-11h00: I. Berkes (Graz University of Technology) 

Title: Extremal limit theorems for heavy tailed processes


Abstract: Extreme value theory goes back to the 1930's and most basic problems in the i.i.d. case have been settled completely. The situation is much less satisfactory in the dependent case, even though many important problems of probability, analysis and number theory lead to extremal problems for dependent processes. In our talk we discuss some open problems in the i.i.d. case such as limit theorems for semistable processes (e.g. the St. Petersburg game) and some unusual results in the theory of modulus trimming. We also discuss applications in analysis, such as extremal results for continued fractions, Weyl's theory of uniform distribution mod 1 and irrational rotations.

             11h00-11h20: Coffee Break 

             11h20-12h00: C.Y. Robert (ISFA, University of Lyon 1) 

Title: Likelihood based inference for high-dimensional extreme value distributions

Abstract: Multivariate extreme value statistical analysis is concerned with observations on several variables which are thought to possess some degree of tail-dependence. In areas such as the modeling of financial and insurance risks, or as the modeling of spatial variables, extreme value models in high dimensions (up to fifty or more) with their statistical inference procedures are needed. In this paper, we consider max-stable models for which the spectral random vectors have absolutely continuous distributions. For random samples with max-stable distributions we provide quasi-explicit analytical expressions of the full likelihoods. When the full likelihood becomes numerically intractable because of a too large dimension, it is however necessary to split the components into subgroups and to consider a composite likelihood approach. For random samples in the max-domain of attraction of a max-stable distribution, two approaches that use simpler likelihoods are possible: (i) a threshold approach that is combined with a censoring scheme, (ii) a block maxima approach that exploits the information on the occurrence times of the componentwise maxima. The asymptotic properties of the estimators are given and the utility of the methods is examined via simulation. The estimators are also compared with those derived from the pairwise composite likelihood method which has been previously proposed in the spatial extreme value literature.

This is a joint work with Alexis Bienvenu.

             12h00-12h40: P. Bertail (University Paris Ouest)      

Title: Empirical processes in survey sampling


Abstract: When the available data is so voluminous that we cannot treat the entire dataset (big data), survey sampling appears as a natural remedy. In particular, as opposed to simple sub-sampling, it may permit to control the efficiency of estimators via the strategic definition of unequal survey weights. It is the main purpose of this talk to review some recent results about variants of empirical process indexed by classes in the context of survey data. Precisely, a functional central limit theorem for general classes of function satisfying some uniform entropy conditions is established when the sample is picked by means of a Poisson survey scheme. This preliminary result is then extended to the case of rejective and conditional Poisson sampling case via conditionnal central limit theorems, and in turn to high entropy survey sampling plans which are close to the rejective design in the sense of the Bounded-Lipschitz distance. Applications to Hadamard- and Frechet differentiable functionals  are also considered. 
Joint work with S. Clémençon and E. Chautru


                Lunch in I.H.P.

    

             14h00-14h40: S Lopes (University of Porto Alegre) 

Title: Theoretical Properties and ML Estimation for Continuous Processes Derived from GLE Solution


Abstract:This work analyzes a class of continuous time process arising from the solution of the generalized Langevin equation (GLE). The main interest is to study this class when the noise process has infinite second moment. We consider the case
where the noise is a symmetric α-stable Lévy process, which can also have infinitee first moment. One goal is to study the dependence structure of the process, but since the autocovariance function is not well-defined we propose to use a different
dependence measure, the so-called codifference. We also propose an estimator for this dependence measure and prove its consistency. A Monte Carlo simulation study is presented showing the generation and the theoretical and empirical cod-
ifference functions for particular processes in this class. Another interest in this work is to estimate the process parameters. The maximum likelihood estimation procedure is proposed to estimate the parameters of the process arising from the
classical Langevin equation, i.e., the Ornstein-Uhlenbeck process, and of the so-called Cosine process. Since the α-stable distribution has closed formula in only three cases, that is, when α ∈ {0.5, 1, 2}, it is necessary to use numerical methods
for the process generation and also for the estimation by maximizing the likelihood function.
This is an undergoing joint work with J. Stein (UFRGS) and A.V. Medino

                14h40-15h20: K. Fokianos (University of Cyprus)     

Title: Consistent testing for pairwise dependence in time series


Abstract: We consider the problem of testing pairwise dependence for stationary time series. We suggest the use of a Box-Ljung type test statistic which is formed after calculating the distance covariance function among pairs of observations. The distance covariance function is a suitable measure for detecting dependencies among data and it is based on the distance between the characteristic function of the joint distribution of the random variables to the product of the marginals. We show that, under the null hypothesis of independence and under mild regularity conditions, the test statistic converges to a normal random variable. The results are complemented by several examples.
This is a joint work with M. Pitsillou.

             15h20-15h40: Coffee Break

             15h40-16h20: E. Rio (University of Versailles-St Quentin)

Title: On Hoeffding's Inequalities
 

Abstract: In this talk, we give some  improvements of the classical inequalities of Hoeffding for sums of independent, bounded from on side, random variables and martingales with bounded increments.

             16h20-17h00: M. Peligrad (University of Cincinnati)     

Title: Random fields, spectral density and empirical spectral distribution


Abstract:In this talk, we will survey some recent results on the empirical eigenvalue distribution of symmetric matrices with dependent entries, selected from regular random fields. It will be pointed out that, in many situations of interest, the limiting spectral measure always exists and depends only on the spectral density of the random field. The strength of the dependence is not important; the field can have long or short range memory and no rate of convergence to zero of the covariances is imposed. We characterize this limit in terms of the Stieltjes transform, via a certain equation involving the spectral density of the field. If the entries of the matrix are square integrable functions of an independent field, the results hold without any other additional assumptions.
The talk is based on joint works with M. Banna and F. Merlevède.