From almost sure local regularity to almost sure Hausdorff dimension for gaussian fields

Erick Herbin; Benjamin Arras; Geoffroy Barruel

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 418-440
  • ISSN: 1292-8100

Abstract

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

How to cite

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Herbin, Erick, Arras, Benjamin, and Barruel, Geoffroy. "From almost sure local regularity to almost sure Hausdorff dimension for gaussian fields." ESAIM: Probability and Statistics 18 (2014): 418-440. <http://eudml.org/doc/274385>.

@article{Herbin2014,
abstract = {Fine regularity of stochastic processes is usually measured in a local way by local Hölder exponents and in a global way by fractal dimensions. In the case of multiparameter Gaussian random fields, Adler proved that these two concepts are connected under the assumption of increment stationarity property. The aim of this paper is to consider the case of Gaussian fields without any stationarity condition. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph in any ball B(t0,ρ) are bounded from above using the local Hölder exponent at t0. We define the deterministic local sub-exponent of Gaussian processes, which allows to obtain an almost sure lower bound for these dimensions. Moreover, the Hausdorff dimensions of the sample path on an open interval are controlled almost surely by the minimum of the local exponents. Then, we apply these generic results to the cases of the set-indexed fractional Brownian motion on RN, the multifractional Brownian motion whose regularity function H is irregular and the generalized Weierstrass function, whose Hausdorff dimensions were unknown so far.},
author = {Herbin, Erick, Arras, Benjamin, Barruel, Geoffroy},
journal = {ESAIM: Probability and Statistics},
keywords = {gaussian processes; Hausdorff dimension; (multi)fractional brownian motion; multiparameter processes; hölder regularity; stationarity; Gaussian random fields; multi-fractional Brownian motion; Hölder regularity},
language = {eng},
pages = {418-440},
publisher = {EDP-Sciences},
title = {From almost sure local regularity to almost sure Hausdorff dimension for gaussian fields},
url = {http://eudml.org/doc/274385},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Herbin, Erick
AU - Arras, Benjamin
AU - Barruel, Geoffroy
TI - From almost sure local regularity to almost sure Hausdorff dimension for gaussian fields
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 418
EP - 440
AB - Fine regularity of stochastic processes is usually measured in a local way by local Hölder exponents and in a global way by fractal dimensions. In the case of multiparameter Gaussian random fields, Adler proved that these two concepts are connected under the assumption of increment stationarity property. The aim of this paper is to consider the case of Gaussian fields without any stationarity condition. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph in any ball B(t0,ρ) are bounded from above using the local Hölder exponent at t0. We define the deterministic local sub-exponent of Gaussian processes, which allows to obtain an almost sure lower bound for these dimensions. Moreover, the Hausdorff dimensions of the sample path on an open interval are controlled almost surely by the minimum of the local exponents. Then, we apply these generic results to the cases of the set-indexed fractional Brownian motion on RN, the multifractional Brownian motion whose regularity function H is irregular and the generalized Weierstrass function, whose Hausdorff dimensions were unknown so far.
LA - eng
KW - gaussian processes; Hausdorff dimension; (multi)fractional brownian motion; multiparameter processes; hölder regularity; stationarity; Gaussian random fields; multi-fractional Brownian motion; Hölder regularity
UR - http://eudml.org/doc/274385
ER -

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