Serge Cohen

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.

The covariance structure of multifractional Brownian motion, with application to long range dependence

Multifractional Brownian motion (mBm) was introduced to overcome certain limitations of the classical fractional Brownian motion (fBm). The major difference between the two processes is that, contrarily to fBm, the almost sure Holder exponent of mBm is allowed to vary along the trajectory, a useful feature when one needs to model processes whose regularity evolves in time, such as Internet traffic or images. Various properties of mBm have been studied in the literature, related to its dimensions or the statistical estimation of its pointwise Holder regularity. However, the covariance structure of mBm has not been investigated so far. We present in this work an explicit formula for this covariance. Since mBm is a zero mean Gaussian process, this provides a full characterization of its stochastic properties. We report on some applications, including the synthesis problem and the long term structure: in particular, we show that the increments of mBm exhibit long range dependence under general conditions.

Gaussian approximation of multivariate Levy processes with applications to simulation of tempered and operator stable processes

Problem of simulation of multivariate Levy processes is investigated. The method based on shot noise series expansions of such processes combined with Gaussian approximation of the remainder is established in full generality. Formulas that can be used for simulation of tempered stable, operator stable and other multivariate processes are obtained.

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.

Random rewards, fractional Brownian local times and stable self-similar processes

We describe a new class of self-similar symmetric

Small and Large Scale Behavior of the Poissonized Telecom Process

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Regularity and identification of Generalized Multifractional Gaussian Processes

-stable processes with stationary increments arising as a large time scale limit in a situation where many users are earning random rewards or incurring random costs. The resulting models are different from the ones studied earlier both in their memory properties and smoothness of the sample paths.

Local self-similarity and the Hausdorff dimension

The stable Telecom process has infinite variance and appears as a limit of renormalized renewal reward processes. We study its Poissonized version where the infinite variance stable measure is replaced by a Poisson point measure. We show that this Poissonized version converges to the stable Telecom process at small scales and to the Gaussian fractional Brownian motion at large scales. This process is therefore locally as well as asymptotically self-similar. The value of the self-similarity parameter at large scales, namely the self-similarity parameter of the limit fractional Brownian motion, depends on the form the Poissonized Telecom process. The Poissonized Telecom process is a Poissonized mixed moving average. We investigate more general Poissonized mixed moving averages as well.

Fractional Fields and Applications

In this article a class of multifractional processes is introduced, called Generalized Multifractional Gaussian Process (GMGP). For such multifractional models, the Hurst exponent of the celebrated Fractional Brownian Motion is replaced by a function, called the multifractional function, which may be irregular. The main aim of this paper is to show how to identify irregular multifractional functions in the setting of GMGP. Examples of discontinuous multifractional functions are also given.