Jacques Istas

Identifying the multifractional function of a Gaussian process

Gaussian processes that are multifractional are studied in this paper. By multifractional processes we mean that they behave locally like a fractional Brownian motion, but the fractional index is no more a constant: it is a function. We introduce estimators of this multifractional function based on discrete observations of one sample path of the process and we study their asymptotical behavior as the mesh decreases to zero.

Identification of filtered white noises

In this paper, a class of Gaussian processes, having locally the same fractal properties as fractional Brownian motion, is studied. Our aim is to give estimators of the relevant parameters of these processes from one sample path. A time dependency of the integrand of the classical Wiener integral, associated with the fractional Brownian motion, is introduced. We show how to identify the asymptotic expansion for high frequencies of these integrands on one sample path. Then, the identification of the first terms of this expansion is used to solve some filtering problems. Furthermore, rates of convergence of the estimators are then given.

Identification of the Hurst Index of a Step Fractional Brownian Motion

We propose a semi-parametric estimator for a piece-wise constant Hurst coefficient of a step fractional Brownian motion (SFBM). For the applications, we want to detect abrupt changes of the Hurst index (which represents long-range correlation) for a Gaussian process with a.s. continuous paths. The previous model of multifractional Brownian motion give a.s. discontinuous paths at change times of the Hurst index. Thus, we first propose a new kind of Fractional Brownian Motion, the SFBM and prove some (Hölder) continuity results. After, we propose an estimator of the piecewise constant Hurst parameter and prove its consistency.

Local self-similarity and the Hausdorff dimension

Abstract Let X be a locally self-similar stochastic process of index 0 H C H − e for all e >0. Then the Hausdorff dimension of the graph of X is a.s. 2− H . To cite this article: A. Benassi et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Fractional Fields and Applications

Foreword.- Contents.- Introduction.- Preliminaries.- Self-similarity.- Asymptotic self-similarity.- Statistics.- Simulations.- A Appendix.- B Appendix.- References.

Ball throwing on spheres

Ball throwing on Euclidean spaces has been considered for some time. A suitable renormalization leads to a fractional Brownian motion as limit object. In this paper, we investigate ball throwing on spheres. A different behavior is exhibited: we still get a Gaussian limit, but it is no longer a fractional Brownian motion. However, the limit is locally self-similar when the self-similarity index H is less than 1 / 2.

Identification d’un processus gaussien multifractionnaire avec des ruptures sur la fonction d’échelle

Resume Nous proposons un modele de processus stochastique gaussien multifractionnaire a trajectoires continues, mais dont la fonction d’echelle presente des discontinuites, i.e. une fonction d’echelle constante par morceaux (SFBM: « Step Fractional Brownian Motion »). Ceci permet de modeliser des phenomenes, a trajectoires continues, comportant des changements abrupts de nature a certains moments. Nous construisons le modele theorique, puis nous proposons un estimateur de la fonction d’echelle du processus en detectant les instants de rupture et en estimant les valeurs de la fonction d’echelle entre les instants de rupture.

Cramèr–Rao bounds for fractional Brownian motions

We obtain Cramer-Rao bounds for parameters estimators of fractional Brownian motions. We point out the differences of behavior whether these processes are standard or not. The key-point of this study relies upon a linear algebra result we prove, exhibiting bounds for elements of inverse of localized matrices.

Self-Similarity and Intermittency

In this paper, we propose a class of stochastic processes having an extended self-similarity property as well as intermittency. These notions are characterized with two parameters, and we propose statistical estimators for them.

Asymptotic Self-Similarity

Self-similarity, as described in the previous chapter, is a global property, and as such, may be to rigid for some applications. Actually, in many situations the self-similarity parameter H is expected to change with time, and in spatial models, with position.