Multifractal analysis of the occupation measures of a kind of stochastic processes
type: Item
instance of: scholarly article
title: Multifractal analysis of the occupation measures of a kind of stochastic processes
zbMATH Open document ID: 0962.60028
DOI: 10.1016/S0304-4149(98)00063-5
author: Xiaoyu Hu
published in: Stochastic Processes and their Applications
publication date: 2001-01-17T00:00:00Z
review text: An \(\alpha \)-sequence (\(0<\alpha <1\)) is a sequence \(\{ a_1,a_2,\ldots \}\) of positive numbers such that \(a_i\searrow 0\), \(\sum _ia_i=1\), \(\sum _i\min \{ a_i,\varepsilon \}\approx \varepsilon ^{1-\alpha}\), \(\sup \{ k:a_k\geq \varepsilon \}\approx \varepsilon ^{-\alpha}\) (\(0<\varepsilon <1\)). Consider the stochastic process \(\{ X(t):t\in [0,1]\}\) defined on the probability space \(\Omega =[0,1]^{\mathbb N}\) with the product Borel \(\sigma \)-algebra and product Lebesgue measure \({\mathcal L}^{\mathbb N}\) by \[ X_\omega (t)=\sum _ia_iI_{\{ 0\leq \omega _i<t\}},\quad t\in [0,1],\quad \omega =(\omega _1,\omega _2,\ldots)\in \Omega . \] Let \(\mu \) be the occupation measure \(\mu \) of \(X\) (\(\mu (\cdot)={\mathcal L}(\{ t:X(t)\in \cdot \})\)). For fixed \(x\in [0,1]\), the local density \(d(\mu ,x)\) of \(\mu \) at \(x\) equals \(\alpha \) almost surely (this follows from some earlier results of the author). In this paper, the set of points where \(d(\mu ,x)\neq \alpha \) is investigated. In particular, bounds for the Hausdorff dimension of the level sets \(\{ x : d (\mu ,x)=\beta \}\), \(\alpha \leq \beta \leq 2\alpha \), are obtained.
reviewed by: Jan Rataj
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 1554301
zbMATH Keywords: local dimension
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: occupation measure
MaRDI profile type: Publication
cites work: Q5563135
cites work: RANDOM RE-ORDERINGS OF INTERVALS COMPLEMENTARY TO A LINEAR SET
cites work: Q3995301
cites work: The multifractal spectrum of statistically self-similar measures
cites work: Fractal measures and their singularities: The characterization of strange sets
cites work: The exact Hausdorff measure for random re-ordering of Cantor set
cites work: The exact packing measure for a random re-ordering of the Cantor set
cites work: The multifractal structure of stable occupation measure
cites work: A multifractal formalism
cites work: Uniform measure results for the image of subsets under Brownian motion
cites work: Q3746589
cites work: The exact hausdorff measure of the zero set of a stable process
description: scientific article; zbMATH DE number 1554301