Sample function behavior of increasing processes of class \(L\)
type: Item
instance of: scholarly article
title: Sample function behavior of increasing processes of class \(L\)
zbMATH Open document ID: 0849.60036
DOI: 10.1007/BF01213685
author: Toshiro Watanabe
publication date: 1996-06-06T00:00:00Z
review text: A stochastic process \(\{X(t), t \geq 0\}\) is said to be a process of class \(L\) with exponent \(H\) if (i) \(X(t)\) is self-similar, that is, for every \(c > 0\), \(\{X (ct)\}\) and \(\{c^H X(t)\}\) have the same finite-dimensional distributions, (ii) \(\{X(t)\}\) has independent increments and (iii) \(X(t)\) is right continuous for \(t \geq 0\) with left limits for \(t > 0\) almost surely. Suppose the process \(\{X(t)\}\) is increasing and of class \(L\) and let \(h(t)\) be an increasing positive function on \((0, \infty)\) with \(h(0+) = 0\) and \(h (\infty) = \infty\). The author gives necessary and sufficient conditions for the existence of a function \(h(t)\) such that \(\lim_{t \to \infty} \sup {X(t) \over h(t)} = C\) or \(\lim_{t \to \infty} \inf {X(t) \over h(t)} = c\) where \(C = c = 1\). He explicitly constructs such \(h(t)\) in case \(h(t)\) exists with \(c = 1\) or \(C = 1\). A criterion to classify functions \(h(t)\) with \(c = 0\) or \(C = 0\) or with \(c = \infty\) or \(C = \infty\) is also given in case \(h(t)\) does not exist with \(c = 1\) or \(C = 1\). Results extend earlier work of \textit{K.-i. Sato} [ibid. 89, No. 3, 285-300 (1991; Zbl 0725.60034)].
Mathematics Subject Classification ID: 60G18
Mathematics Subject Classification ID: 60J99
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 878010
zbMATH Keywords: self-similar
zbMATH Keywords: finite-dimensional distributions
zbMATH Keywords: increasing positive function
reviewed by: B. L. S. Prakasa Rao
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 878010