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From almost sure local regularity to almost sure Hausdorffdimension for Gaussian fields

Published online by Cambridge University Press:  08 October 2014

Erick Herbin,
Benjamin Arras and
Geoffroy Barruel
Erick Herbin
Affiliation:
Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry, France. erick.herbin@ecp.fr; benjamin.arras@ecp.fr; geoffroy.barruel@student.ecp.fr
Benjamin Arras
Affiliation:
Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry, France. erick.herbin@ecp.fr; benjamin.arras@ecp.fr; geoffroy.barruel@student.ecp.fr
Geoffroy Barruel
Affiliation:
Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry, France. erick.herbin@ecp.fr; benjamin.arras@ecp.fr; geoffroy.barruel@student.ecp.fr

Abstract

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Fine regularity of stochastic processes is usually measured in a local way by localHölder exponents and in a global way by fractal dimensions. In the case of multiparameterGaussian random fields, Adler proved that these two concepts are connected under theassumption of increment stationarity property. The aim of this paper is to consider thecase of Gaussian fields without any stationarity condition. More precisely, we prove thatalmost surely the Hausdorff dimensions of the range and the graph in any ballB(t0,ρ)are bounded from above using the local Hölder exponent at t0. We definethe deterministic local sub-exponent of Gaussian processes, which allows to obtain analmost sure lower bound for these dimensions. Moreover, the Hausdorff dimensions of thesample path on an open interval are controlled almost surely by the minimum of the localexponents. Then, we apply these generic results to the cases of the set-indexed fractionalBrownian motion on RN, the multifractionalBrownian motion whose regularity function H is irregular and the generalized Weierstrassfunction, whose Hausdorff dimensions were unknown so far.

Keywords

Gaussian processesHausdorff dimension(multi)fractional Brownian motionmultiparameter processesHölder regularitystationarity

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014, pp. 418 - 440
Copyright
© EDP Sciences, SMAI 2014

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