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. ...

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...

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...

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...

Ivan NOURDIN. Université Paris 6. 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

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...

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...

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...

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...

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 :...

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 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...

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) ...

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...

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...

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,...

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 :...