Fractional Brownian motion and packing dimension
type: Item
instance of: scholarly article
title: Fractional Brownian motion and packing dimension
zbMATH Open document ID: 0860.60064
DOI: 10.1007/BF02214076
author: Michel Talagrand
author: Y. Xiao
published in: Journal of Theoretical Probability
publication date: 1997-04-21T00:00:00Z
review text: Let \(X\) be a fractional Brownian motion of index \(\alpha\in]0,1[\), from \(\mathbb{R}\) into \(\mathbb{R}^d\), with \(1>\alpha d\), and let \(\text{Dim }F\) denote the packing dimension of any set \(F\) in \(\mathbb{R}^N\). A compact set \(E\) of \([0,1]\) is constructed, such that \(\text{Dim }X(E)<{1\over\alpha}\text{ Dim }E\); this invalidates a general conjecture on \(\text{Dim }X(E)\). On the other hand, the following general lower bound is given: for any compact set \(E\) in \(\mathbb{R}\), \[ \text{Dim }X(E)\geq {\text{Dim }E\times d\over \alpha d+\text{Dim }E\times(1-\alpha d)}, \] almost surely. And this bound is optimal, being attained for the peculiar \(E\) constructed above.
reviewed by: Jacques Franchi
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 934228
zbMATH Keywords: packing dimension
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: image
MaRDI profile type: Publication
cites work: Q3995301
cites work: Q3687398
cites work: Q5562405
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full work available at URL: https://doi.org/10.1007/bf02214076
OpenAlex ID: W1981527857
description: scientific article; zbMATH DE number 934228