Jean Jacod. Université Paris6. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1265816883468 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 51 mn

Limit theorems and applications (SAMSOS, 2008)

01 - Estimating the Degree of Activity of jumps in High Frequency Data - Jean JACOD

In this paper, we give estimates of ideal or minimal distances between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary martingale difference sequences or stationary sequences satisfying projective criteria. Applications to functions of linear processes and to functions of expanding maps of the interval are given. This is a joint paper with J. Dedecker (Paris 6) and F. Merlevède (Paris 6). Emmanuel RIO. Université de Versailles. Ecouter l'intervention : Bande son disponible au format mp3 Durée : 44 mn

02 - Rates of convergence for minimal distances in the central limit theorem under projective criteria - Emmanuel RIO

We will show taht one can combine Malliavin calculus with Stein's method, in order to derive explicit bounds in the Gaussian and Gamma approximations of arbitrary regular functionals of a given Gaussian field (here, the notion of regularity is in the sense of Malliavin derivability). When applied to random variables belonging to a fixed Wiener chaos, our approach generalizes, refines proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We shall discuss some connections with the classic method of moments and cumulants. As an application, we deduce explicit Berry-Esseen bounds in the Breuer-Major Central limit theorem for subordinated functionals of a fractional Brownian motion. This talk is based on joint works with I. Nourdin (Paris VI). Giovanni PECCATI. Université de Paris 6. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750005329 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 41 mn

03 - Stein's method and weak convergence on Wiener space - Giovanni PECCATI

We investigate the asymptotic behavior of a particular family of weighted sums of independent standardized random variables with uniformly bounded third moments. We prove that the empirical CDF of the resulting partial sums converges almost surely to the normal CDF. It allows us to deduce the almost sure uniform convergence of empirical distribution of the empirical periodogram as well as the almost sure uniform convergence of spectral distribution of symmetric circulant random matrices. In the special case of trigonometric weights, we also establish a central limit theorem and a large deviation principle. It is a joint workwith W. Bryc. Bernard BERCU Université de Bordeaux 1 Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750057287 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 35 mn

04 - Asymptotic results for empirical measures of weighted sums of independant random variables- Bernard BERCU

We study the weak convergence (in the high-frequency limit) of the frequency components associated with Gaussian-subordinated, spherical and isotropic random fields. In particular, we provide conditions for asymptotic Gaussianity and we establish a new connection with random walks on the the dual of SO(3), which mirrors analogous results previously established for fields defined on Abelian groups. Our work is motivated by applications to cosmological data analysis, and specifically by the probabilistic modelling and the statistical nalysis of the Cosmic Microwave Background radiation, which is currently at the frontier of physical research. To obtain our main results, we prove several fine estimates involving convolutions of the so-called Clebsch-Gordan coefficients (which are elements of unitary matrices connecting reducible representations of SO(3)); this allows to intepret most of our asymptotic conditions in terms of coupling of angular momenta in a quantum mechanical system. Part of the proofs are based on recently established criteria for the weak convergence of multiple Wiener-Itô integrals. This is a joint paper by Domenico Marinucci (Rome "Tor Vergata") and Giovanni Peccati (Paris VI). Domenico MARINUCCI Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750104939 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 57 mn

05 - Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields - Domenico MARINUCCI

Ivan NOURDIN. Université Paris 6. Ecouter l'intervention : Bande son disponible au format mp3 Durée : 47 mn

06 - Weighted power variations of fractional and iterated Brownian motions - Ivan NOURDIN

We study a group of related problems: the extent to which presence of regular variation of the tail of certain $sigma$-finite measures at the output of a linear filter determines the corresponding regular variation of a measure at the input to the filter. This turns out to be related to presence of a particular cancellation property in $sigma$-finite measures, which, in turn, is related to uniqueness of solutions of certain functional equations. The techniques we develop are applied to weighted sums of iid random variables, to products of independent random variables, and to stochastic integrals with respect to Lévy motions. Joint work with Martin Jacobsen, Thomas Mikosch and Jan Rosinski. Gennady SAMORODNITSKY. Cornell University. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750230504 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 47 mn

01 - Inverse problems for regular variation, linear filters, functional equations and a cancellation property for $sigma$-finite measures - Gennady SAMORODNITSKY

Philippe SOULIER Université Paris 10 Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750174352 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 33 mn

07 - On the existence of some ARCH($infty$) processes - Philippe SOULIER

In the work (Bender, T. Sottinen, and E. Valkeila (2006)) we show that it is possible to extend the classical Black & Scholes hedging for a class of models, where the quadratic variation is identical to the Black & Scholes model. Dzhaparidze and Spreij show in (K. Dzhaparidze, and P. Spreij (1994)), that the periodogram constructed from the process estimates the quadratic variation in the semimartingale setting.We show that the periodogram estimates the quadratic variation for the mixed Brownian fractional Brownian motion, too.The talk is based on joint with Ehsan Azmoodeh. Esko VALKEILA. Helsinky University of Technology. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750279333 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 41 mn

02 - An estimator for the quadratic variation of mixed Brownian fractional - Esko VALKEILA

We prove a central limit theorem for linear triangular arrays under weak dependence conditions [1,3,4]. Our result is then applied to the study of dependent random variables sampled by a $Z$-valued transient random walk. This extends the results obtained by Guillotin-Plantard & Schneider [2]. An application to parametric estimation by random sampling is also provided. References: [1] Dedecker J., Doukhan P., Lang G., Leon J.R., Louhichi S. and Prieur C. (2007). Weak dependence: With Examples and Applications. Lect. notes in Stat. 190. Springer, XIV. [2] N. Guillotin-Plantard and D. Schneider (2003). Limit theorems for sampled dynamical systems. Stochastic and Dynamics 3, 4, p. 477-497. [3] M. Peligrad and S. Utev (1997). Central limit theorem for linear processes. Ann. Probab. 25, 1, p. 443-456. [4] S. A. Utev (1991). Sums of random variables with $varphi$-mixing. Siberian Advances in Mathematics 1, 3, p. 124-155. Clémentine PRIEUR. Université de Toulouse 1. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750339872 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 33 mn

03 - Central limit theorem for sampled sums of dependent random variables - Clémentine PRIEUR

We introduce a new modification of Sentana's (1995) Quadratic ARCH (QARCH), the Linear ARCH (LARCH) (Giraitis et al., 2000, 2004) and the bilinear models (Giraitis and Surgailis, 2002), which can combine the following properties: (a.1) conditional heteroskedasticity (a.2) long memory (a.3) the leverage effect (a.4) strict positivity of volatility (a.5) Lévy-stable limit behavior of partial sums of squares Sentana's QARCH model is known for properties (a.1), (a.3), (a.4), and the LARCH model for (a.1), (a.2), (a.3). Property (a.5) is new. References: [1] Giraitis, L., Robinson, P.M., Surgailis, D. (2000) A model for long memory conditional heteroscedasticity, Ann. Appl. Probab. 10, 1002--1024. [2] Giraitis, L., Surgailis, D. (2002) ARCH-type bilinear models with double long memory, Stoch. Process. Appl. 100, 275--300. [3] Giraitis, L., Leipus, R., Robinson, P.M., Surgailis, D. (2004) LARCH, leverage and long memory, J. Financial Econometrics 2, 177--210. [4] Sentana, E. (1995) Quadratic ARCH models, Rev. Econ. Stud. 3, 77--102. Donatas SURGAILIS. Academy of Sciences, Lithuania. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750384173 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 37 mn

04 - Lévy-stable behavior of squares - Donatas SURGAILIS

05 - Weak dependence, Models and Some Applications - Paul DOUKHAN

Nous définissons une classe de processus multifractaux en intégrant une cascades multiplicative stationnaire contre un mouvement brownien fractionnaire. Les propriétés de scaling sont étudiées ainsi que le formalisme multifractal associé. This talk is based on a joint work with P.Abry, P.Chainais et V.Pipiras. Laure COUTIN. Université Paris 5. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750545594 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 46 mn

06 - Multifractional random walk as fractional integral - Laure COUTIN

Contraction rates of posterior distributions on nonparametric models are derived for Gaussian process priors. We show that the convergence rate depends on the small ball probabilities of the Gaussian process and on the position of the true parameter relative to the reproducing kernel Hilbert space of the Gaussian process. Explicit examples are given for various statistical settings, including density estimation, nonparametric regression, and classification. We also discuss how rescaling of the prior process affects the contraction rates and how random rescaling can yield rate-adaptive procedures. This is based on joint work with Aad van der Vaart. Harry VAN ZANTEN. Vrije Universiteit. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750607556 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 34 mn

08 - Contraction rates of posteriors based on Gaussian - Harry VAN ZANTEN

The paper develops a limit theory for the quadratic form $Q_{n,X}$ in linear random variables $X_1, ldots, X_n$ which can be used to derive the asymptotic normality of various semiparametric, kernel, window and other estimators converging at a rate which is not necessarily $n^{1/2}$. The theory covers practically all forms of linear serial dependence including long, short and negative memory, and provides conditions which can be readily verified thus eliminating the need to develop technical arguments for special cases. This is accomplished by establishing a general CLT for $Q_{n,X}$ with normalization $(var[Q_{n,X}])^{1/2}$ assuming only $2+delta$ finite moments. Previous results for forms in dependent variables allowed only normalization with $n^{1/2}$ and required at least four finite moments. Our technique uses approximations of $Q_{n, X}$ by a form $Q_{n, Z}$ in i.i.d. errors $Z_1, ldots, Z_n$. We develop sharp bounds for these approximations which in some cases are faster by the factor $n^{1/2}$ compared to the existing results. Liudas GIRAITIS Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750657882 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 51 mn

01 - Approximations and limit theory for quadratic forms of linear processes - Liudas GIRAITIS

L'étude des processus stochastiques est un domaine mathématique qui connait un réel développement aussi bien d'un point de vue théorique que du coté des applications. Le but de cette conférence est de proposer un panorama des résultats nouveaux sur les théorèmes limites pour les processus stochastiques (dont les techniques mise en oeuvre pourront aussi bien reposer sur le calcul stochastique ou sur différentes notions de dépendance) et sur leurs applications aux statistiques. Les supports de présentation sont disponibles sur l'Espace pédagogique interactif (http://epi.univ-paris1.fr/samos-limit-theorems-application). Recommandé à : étudiant de la discipline, chercheur - Catégorie : conférences - Année de réalisation : 2008

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Limit theorems and applications (SAMSOS, 2008)

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