293 research outputs found
Convergence to Weighted Fractional Brownian Sheets
We define weighted fractional Brownian sheets, which are a class of Gaussian
random fields with four parameters that include fractional Brownian sheets as
special cases, and we give some of their properties. We show that for certain
values of the parameters the weighted fractional Brownian sheets are obtained
as limits in law of occupation time fluctuations of a stochastic particle
model. In contrast with some known approximations of fractional Brownian sheets
which use a kernel in a Volterra type integral representation of fractional
Brownian motion with respect to ordinary Brownian motion, our approximation
does not make use of a kernel
Image sets of fractional Brownian sheets
Let be an -fractional Brownian
sheet with Hurst index . The main objective of
the present paper is to study the Hausdorff dimension of the image sets
, and , in the dimension
case . Following the seminal work of
Kaufman (1989), we establish uniform dimensional properties on , answering
questions raised by Khoshnevisan et al (2006) and Wu and Xiao (2009).
For the purpose of this work, we introduce a refinement of the sectorial
local-nondeterminism property which can be of independent interest to the study
of other fine properties of fractional Brownian sheets.Comment: 14 pages, 1 figur
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
Joint continuity of the local times of fractional Brownian sheets
Let be an -fractional Brownian
sheet with index defined by
where
are independent copies of a real-valued fractional Brownian
sheet . We prove that if , then the
local times of are jointly continuous. This verifies a conjecture of Xiao
and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp
local and global H\"{o}lder conditions for the local times of . These
results are applied to study analytic and geometric properties of the sample
paths of .Comment: Published in at http://dx.doi.org/10.1214/07-AIHP131 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
High-resolution product quantization for Gaussian processes under sup-norm distortion
We derive high-resolution upper bounds for optimal product quantization of
pathwise contionuous Gaussian processes respective to the supremum norm on
[0,T]^d. Moreover, we describe a product quantization design which attains this
bound. This is achieved under very general assumptions on random series
expansions of the process. It turns out that product quantization is
asymptotically only slightly worse than optimal functional quantization. The
results are applied e.g. to fractional Brownian sheets and the
Ornstein-Uhlenbeck process.Comment: Version publi\'ee dans la revue Bernoulli, 13(3), 653-67
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