Maximal inequalities for additive processes
type: Item
instance of: scholarly article
title: Maximal inequalities for additive processes
zbMATH Open document ID: 1268.60059
author: Michael J. Klass
author: Ming Yang
published in: Journal of Theoretical Probability
publication date: 2013-01-11T00:00:00Z
review text: A process \(X_{t}\), \(t\geq 0\), with independent increments, right-continuous with left-limits (rcll) paths, and values in \(\mathbb{R}^{d}\) is called additive if \(X_{t}\) is continuous in probability and \(X_{0}=0\). A nondecreasing and right-continuous function \(\phi (\cdot )\) with \(\phi (0)=0\), \(\phi (x)>0\), \(x>0\), is called moderate if there exist two constants \(\gamma \), \(\theta \in (0,\infty )\) such that \(\phi (x_{2})/\phi (x_{1})\leq \gamma (x_{2}/x_{1})^{\theta }\) whenever \(0<x_{1}<x_{2}<\infty \). As one of a number of results, the authors prove the following. Let \(X_{t}\) be an arbitrary additive process taking values in \(\mathbb{R}^{d}\), and let \(\phi (\cdot )\) be a moderate function. Consider \(X_{t}^{\ast }=\sup_{0\leq s\leq t}\left\| X_{s}\right\| \), where \(\left\| \cdot \right\| \) stands for Euclidean norm. A function \(a_{\phi }(\cdot )\) can be constructed in terms of the characteristics of \(X_{t}\) such that \(c_{1}\leq E\phi (X_{t}^{\ast })/Ea_{\phi }(t)\leq c_{2}\) holds uniformly for all \(t>0\), where \(0<c_{1}\leq c_{2}<\infty \) are two constants depending on \(\phi \) only.
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 6124634
zbMATH Keywords: additive processes
zbMATH Keywords: Lévy processes
zbMATH Keywords: maximal inequalities
zbMATH Keywords: stopping times
zbMATH Keywords: random walks
zbMATH Keywords: moderate functions
zbMATH Keywords: sums of independent random variables
reviewed by: Qihe Tang
MaRDI profile type: Publication
full work available at URL: s10959-011-0357-4
OpenAlex ID: W2080544237
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DOI: 10.1007/S10959-011-0357-4
description: scientific article; zbMATH DE number 6124634