Real harmonizable multifractional Lévy motions

In this paper, the class of real harmonizable multifractional Lévy motions (in short RHMLMs) is introduced. This class is a generalization of the multifractional Brownian motion (in short MBM) and of the class of real harmonizable fractional Lévy motions. One of its main interest is that it contains some non-Gaussian fields which share many properties with the MBM. RHMLMs have locally Hölder sample paths and their Hölder exponent is allowed to vary along the trajectory. Moreover these fields are locally asymptotically self-similar. The multifractional function can be estimated with the localized generalized quadratic variations

Time-changed extremal process as a random sup measure

A functional limit theorem for the partial maxima of a long memory stable sequence produces a limiting process that can be described as a $\beta$-power time change in the classical Fr\'echet extremal process, for $\beta$ in a subinterval of the unit interval. Any such power time change in the extremal process for $0<\beta<1$ produces a process with stationary max-increments. This deceptively simple time change hides the much more delicate structure of the resulting process as a self-affine random sup measure. We uncover this structure and show that in a certain range of the parameters this random measure arises as a limit of the partial maxima of the same long memory stable sequence, but in a different space. These results open a way to construct a whole new class of self-similar Fr\'echet processes with stationary max-increments.Comment: Published at http://dx.doi.org/10.3150/15-BEJ717 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A general framework for simulation of fractional fields

International audienceBesides fractional Brownian motion most non-Gaussian fractional fields are obtained by integration of deterministic kernels with respect to a random infinitely divisible measure. In this paper, generalized shot noise series are used to obtain approximations of most of these fractional fields, including linear and harmonizable fractional stable fields. Almost sure and $L^r$-norm rates of convergence, relying on asymptotic developments of the deterministic kernels, are presented as a consequence of an approximation result concerning series of symmetric random variables. When the control measure is infinite, normal approximation has to be used as a complement. The general framework is illustrated by simulations of classical fractional fields

Monte Carlo methods for light propagation in biological tissues

Light propagation in turbid media is driven by the equation of radiative transfer. We give a formal probabilistic representation of its solution in the framework of biological tissues and we implement algorithms based on Monte Carlo methods in order to estimate the quantity of light that is received by a homogeneous tissue when emitted by an optic fiber. A variance reduction method is studied and implemented, as well as a Markov chain Monte Carlo method based on the Metropolis–Hastings algorithm. The resulting estimating methods are then compared to the so-called Wang–Prahl (or Wang) method. Finally, the formal representation allows to derive a non-linear optimization algorithm close to Levenberg–Marquardt that is used for the estimation of the scattering and absorption coefficients of the tissue from measurement

LAN property for some fractional type Brownian motion

International audienceWe study asymptotic expansion of the likelihood of a certain class of Gaussian processes characterized by their spectral density $f_\theta$. We consider the case where f_\theta\PAR{x} \sim_{x\to 0} \ABS{x}^{-\al(\theta)}L_\theta(x) with $L_\theta$ a slowly varying function and \al\PAR{\theta}\in (-\infty,1). We prove LAN property for these models which include in particular fractional Brownian motion %$B^\alpha_t,\: \alpha \geq 1/2$ or ARFIMA processes

Modulus of continuity of some conditionally sub-Gaussian fields, application to stable random fields

International audienceIn this paper we study modulus of continuity and rate of convergence of series of conditionally sub-Gaussian random fields. This framework includes both classical series representations of Gaussian fields and LePage series representations of stable fields. We enlighten their anisotropic properties by using an adapted quasi-metric instead of the classical Euclidean norm. We specify our assumptions in the case of shot noise series where arrival times of a Poisson process are involved. This allows us to state unified results for harmonizable (multi)operator scaling stable random fields through their LePage series representation, as well as to study sample path properties of their multistable analogous