Limit theorems and applications (SAMSOS, 2008)

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

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

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

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

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

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

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

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

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

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