Local properties of a multifractional stable field
type: Item
instance of: scholarly article
title: Local properties of a multifractional stable field
zbMATH Open document ID: 1285.60045
DOI: 10.1090/S0094-9000-2013-00882-5
published in: Theory of Probability and Mathematical Statistics
publication date: 2013-09-17T00:00:00Z
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60G22
zbMATH DE Number: 6208806
zbMATH Keywords: stable process
zbMATH Keywords: harmonizable process
zbMATH Keywords: multifractionality
zbMATH Keywords: local time
zbMATH Keywords: local nondeterminism
author: G. M. Shevchenko
MaRDI profile type: Publication
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full work available at URL: s0094-9000-2013-00882-5
OpenAlex ID: W2048682802
review text: The author considers anisotropic harmonizable multifractional stable \((N,d)\)-fields of the form NEWLINE\[NEWLINEZ(t)=\Re\int_{\mathbb R^N}\prod_{k=1}^N\frac{e^{i\,t_kx_k}-1}{|x_k|^{1/\alpha-H_k(t)}}\,M(dx)\quad\text{ for }t\in\mathbb R^NNEWLINE\]NEWLINE with Hölder-continuous Hurst function \(H=(H_1,\dots,H_N)\) in \((0,1)^N\) and \(M=(m_1,\dots,m_d)\) for independent complex rotationally invariant symmetric \(\alpha\)-stable measures \(m_1,\dots,m_d\) on \(\mathbb R^N\) for some \(\alpha\in(1,2]\). Continuity of such fields is shown and the author gives upper and lower bounds for the increments of the field. Further, existence and regularity of the local time of the field is proven. In the Gaussian case \(\alpha=2\), a certain variant of the field is shown to have the property of sectorial local nondeterminism. As a consequence, this implies joint continuity of the local time and enables to compute the Hausdorff dimension of the image of the field.
description: scientific article; zbMATH DE number 6208806