116 research outputs found
The covariance structure of multifractional Brownian motion, with application to long range dependence
International audienceMultifractional 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 \ho 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 already been investigated in the literature, related for instance to its dimensions or the statistical estimation of its pointwise \ho 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, such a formula provides a full characterization of its stochastic properties. We briefly 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
Invariance principle, multifractional Gaussian processes and long-range dependence
This paper is devoted to establish an invariance principle where the limit
process is a multifractional Gaussian process with a multifractional function
which takes its values in . 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.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP127 the Annales de
l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques
(http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Fractional Brownian fields, duality, and martingales
In this paper the whole family of fractional Brownian motions is constructed
as a single Gaussian field indexed by time and the Hurst index simultaneously.
The field has a simple covariance structure and it is related to two
generalizations of fractional Brownian motion known as multifractional Brownian
motions. A mistake common to the existing literature regarding multifractional
Brownian motions is pointed out and corrected. The Gaussian field, due to
inherited ``duality'', reveals a new way of constructing martingales associated
with the odd and even part of a fractional Brownian motion and therefore of the
fractional Brownian motion. The existence of those martingales and their
stochastic representations is the first step to the study of natural wavelet
expansions associated to those processes in the spirit of our earlier work on a
construction of natural wavelets associated to Gaussian-Markov processes.Comment: Published at http://dx.doi.org/10.1214/074921706000000770 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
From N-parameter fractional Brownian motions to N-parameter multifractional Brownian motions
Multifractional Brownian motion is an extension of the well-known fractional
Brownian motion where the Holder regularity is allowed to vary along the paths.
In this paper, two kind of multi-parameter extensions of mBm are studied: one
is isotropic while the other is not. For each of these processes, a moving
average representation, a harmonizable representation, and the covariance
structure are given. The Holder regularity is then studied. In particular, the
case of an irregular exponent function H is investigated. In this situation,
the almost sure pointwise and local Holder exponents of the multi-parameter mBm
are proved to be equal to the correspondent exponents of H. Eventually, a local
asymptotic self-similarity property is proved. The limit process can be another
process than fBm.Comment: 36 page
Some sample path properties of multifractional Brownian motion
The geometry of the multifractional Brownian motion (mBm) is known to present
a complex and surprising form when the Hurst function is greatly irregular.
Nevertheless, most of the literature devoted to the subject considers
sufficiently smooth cases which lead to sample paths locally similar to a
fractional Brownian motion (fBm). The main goal of this paper is therefore to
extend these results to a more general frame and consider any type of
continuous Hurst function. More specifically, we mainly focus on obtaining a
complete characterization of the pointwise H\"older regularity of the sample
paths, and the Box and Hausdorff dimensions of the graph. These results, which
are somehow unusual for a Gaussian process, are illustrated by several
examples, presenting in this way different aspects of the geometry of the mBm
with irregular Hurst functionsComment: 33 pages, 2 figure
Local times of multifractional Brownian sheets
Denote by a function in
with values in . Let
be an
-multifractional Brownian sheet (mfBs) with Hurst functional .
Under some regularity conditions on the function , we prove the
existence, joint continuity and the H\"{o}lder regularity of the local times of
. We also determine the Hausdorff dimensions of the level sets
of . Our results extend the corresponding results for
fractional Brownian sheets and multifractional Brownian motion to
multifractional Brownian sheets.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ126 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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