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

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

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

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

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

We propose a method for numerical approximation of Reflected Backward Stochastic Differential Equations. Is based in the approximation for the Brownian motion by a simple random walk. We prove a weak convergence. This talk is based on joint work with Miguel Martinez and Jaime San Martin. Soledad TORRES. Universidad de Valparaiso. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750701378 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 44 mn

We study generalized random fields which arise as rescaling limits of spatial configurations of uniformly scattered random balls as the mean radius of the balls tends to $0$ or infinity. Assuming that the radius distribution has a power law behavior, we prove that the centered and renormalized random balls field admits a limit with strong spatial dependence. In particular, our approach provides a unified framework to obtain all self-similar, stationary and isotropic Gaussian fields. In addition to investigating stationarity and self-similarity properties, we give $L^2$-representations of the limiting generalized random fields viewed as continuous random linear functionals. Joint work with A. Estrade (Paris 5) and Ingemar Kaj (Uppsala University) Hermine BIERME Université René Descartes Paris 5 Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750733740 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 41 mn

We propose the concept of Local Continuity that is somewhat related to directional continuity. DEFINITION: Let X and Y be, say, metric spaces. A function f from X to Y is locally continuous at point x in X if one can find an open set U(x) such that (i) x belongs to the closure of U(x), (ii) if x(n) converges to x in U(x) then f(x(n)) converges to f(x) in Y. The set U(x), the local continuity set of f at x, that tells the direction of continuity. If U(x) can be chosen to contain x then f is continuous at x. The concept was conceived during our study [Bender, C., Sottinen, T., and Valkeila, E. (2007): Pricing by hedging and no-arbitrage beyond semimartingales (under revision for Finance and Stochastics)] where we considered non-semimartingale pricing models that have non-trivial quadratic variation and a certain "small-ball property". It turned out that in these models one cannot do arbitrage with strategies that are continuous in terms of the spot and some other economic factors such as the running minimum and maximum of the stock. Unfortunately, this result does not extend to even simple strategies, when stopping times are involved. The reason is obvious: Stopping times are typically not continuous. However, local continuity turns out to be just what we need to prove our theorems, and the author is not aware of any reasonable stopping times that are not locally continuous. The talk is based on an ongoing joint work with C. Bender (Technical University of Braunschweig), D. Gasbarra (University of Helsinki), and E. Valkeila (Helsinki University of Technology). Tommi SOTTINEN. Reykjavik University. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750775297 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 49 mn

A popular Bayesian nonparametric approach to survival analysis consists in modeling hazard rates as kernel mixtures driven by a completely random measure. A comprehensive analysis of the asymptotic behaviour of such models is provided. Consistency of the posterior distribution is investigated and central limit theorems for both linear and quadratic functionals of the posterior hazard rate are derived. The general results are then specialized to various specific kernels and mixing measures, thus yielding consistency under minimal conditions and neat central limit theorems for the distribution of functionals. Joint work with P. De Blasi and G. Peccati. Igor PRUNSTER. University of Turin. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750819712 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 46 mn

We establish an invariance principle where the limit process is a multifractional Gaussian process with a multifractional function which takes its values in (1/2, 1). Some properties, such as regularity and local self-similarity, of this process are studied. Moreover the limit process is compared to the multifractional Brownian motion. Renaud MARTY. Université Nancy1. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750851448 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 32 mn