On the fractal nature of increments of \(l_p\)-valued Gaussian processes
type: Item
instance of: scholarly article
title: On the fractal nature of increments of \(l_p\)-valued Gaussian processes
zbMATH Open document ID: 0940.60050
DOI: 10.1016/S0304-4149(97)00063-X
published in: Stochastic Processes and their Applications
publication date: 2000-03-01T00:00:00Z
review text: The author studies the set of points where the oscillation of \(l_p\)-valued Gaussian processes occur infinitely often. Namely, let \(Y(t)=(X_k(t))\) be a sequence of independent Gaussian processes with \(EX_k(t)=0\) and let \(\sigma ^2_k(h)=E(X_k( t+h) -X_k(t))^2\). Further, put \[ \widetilde {\sigma }(p,h)=\begin{cases} \bigl (\sum _{k=1}^\infty \sigma ^s_k(h)\bigr)^{1/s},\;s= \frac {2p}{2-p}, & \text{if} 1\leq p<2;\\ \max _k \sigma _k(h), & \text{if} p\geq 2. \end{cases} \] Then it is shown that under suitable conditions the set \[ \Biggl\{ t\in [0,1];\;\limsup _{h\to 0} \frac {\|Y(t+h)-Y(t)\|_{l_p}}{\widetilde {\sigma }(p,h)(2\log h^{-1})^{1/2}} \geq \alpha \Biggr\}, \quad \alpha \in [0,1], \] is random fractal with the Hausdorff dimension \(1-\alpha ^2\) almost surely. As a consequence the author obtains the estimate of Hausdorff dimensions in the following special cases: \(Y(t)=(c_kX_k(y))\), where \(X_k(t)\) are independent copies of a centered Gaussian process, \((c_k)\in l_p\); \(Y(t)\) is a sequence of independent Ornstein-Uhlenbeck processes.
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 1409136
zbMATH Keywords: fractal nature
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Ornstein-Uhlenbeck processes
zbMATH Keywords: Gaussian processes
author: Lixin Zhang
MaRDI profile type: Publication
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full work available at URL: s0304-4149(97)00063-x
OpenAlex ID: W1980299817
description: scientific article; zbMATH DE number 1409136