Defining fractals in a probability space
type: Item
instance of: scholarly article
title: Defining fractals in a probability space
zbMATH Open document ID: 0797.60001
author: Chaoshou Dai
author: S. James Taylor
published in: Illinois Journal of Mathematics
publication date: 1994-10-10T00:00:00Z
review text: Starting with a fixed stochastic process \(X_ 1,X_ 2,\dots\) taking values in a countable set \(S\), the cylinder sets of order \(n\) consist of points \(w\in \Omega\), where \(X_ i(w)= S_ i\), \(1\leq i\leq n\). The class of all cylinder sets can be used for covering a set \(A\subset \Omega\), or for packing \(A\). This leads to a definition of Hausdorff measure or packing measure with respect to a monotone function \(\psi\). In turn this leads to three dimension indices, \(0\leq \dim A\leq\text{Dim }A\leq \Delta(A)\leq 1\), for any set \(A\subset\Omega\). The condition \(\dim A=\text{Dim }A\) is a possible condition defining a fractal set \(A\); as an example, exceptional sets for a simple random walk are shown to be fractal.
Mathematics Subject Classification ID: 60A10
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 599244
zbMATH Keywords: probability space
zbMATH Keywords: dimension
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: packing measure
zbMATH Keywords: random walk
zbMATH Keywords: fractal
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 599244