Almost sure oscillation of certain random processes
type: Item
instance of: scholarly article
title: Almost sure oscillation of certain random processes
zbMATH Open document ID: 0885.60018
DOI: 10.2307/3318523
author: Jean-Marc Azaïs
author: Mario Wschebor
published in: Bernoulli
publication date: 1998-04-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bj/1178291722
review text: For a real-valued random process \(\{X_t: t\in R\}\) and an appropriate normalizing function \(a(\varepsilon)\), \(\varepsilon\geq 0\), define the process \(Z_\varepsilon (t) = (X_{t+ \varepsilon} -X_t)/a (\varepsilon)\), \(t\geq 0\), and the random measure \(\mu(\varepsilon) = {1\over \lambda(I)} \lambda (\{t\in I: Z_\varepsilon(t) \in B\})\), \(B\) Borel set in \(R\), where \(\lambda\) is the Lebesgue measure. The authors show that for a large family of processes the random measures \(\mu_\varepsilon\) converge weakly as \(\varepsilon\to 0\), for almost all elementary events (almost sure) to a nondegenerate measure \(\mu^*\). The results of this kind are proven for the following classes: (1) a class of Gaussian processes including fractional Brownian motion, stationary processes with certain local behaviour, (2) processes with independent increments and symmetric stable law, (3) continuous martingales satisfying some regularity conditions.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60G99
zbMATH DE Number: 947103
zbMATH Keywords: almost sure oscillation
zbMATH Keywords: Gaussian process
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: martingales
zbMATH Keywords: occupation measure
zbMATH Keywords: stable processes
zbMATH Keywords: symmetric stable law
reviewed by: Władysław Szczotka
MaRDI profile type: Publication
OpenAlex ID: W4229628906
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description: scientific article; zbMATH DE number 947103