Self-similar random measures are locally scale invariant
type: Item
instance of: scholarly article
title: Self-similar random measures are locally scale invariant
zbMATH Open document ID: 0794.60045
DOI: 10.1007/BF01192964
publication date: 1994-08-15T00:00:00Z
review text: In an earlier paper the first author and \textit{U. Zähle} [Math. Nachr. 149, 285-302 (1990; Zbl 0719.60001)] have proved the existence of a fractional tangent measure at the typical point of a self-similar random measure \(\Phi\) under rather special technical assumptions. In the present paper we remove the most restrictive one. Here we suppose the open set condition for the similarities, a constant positive lower bound for the random contraction ratios, and vanishing \(\Phi\) on the boundary of the open set with probability 1. The tangent measure is \(D\)-scale-invariant, where \(D\) is the similarity dimension of \(\Phi\). Moreover, we approximate the tangential distribution by means of \(\Phi\) and use this in order to prove that the Hausdorff dimension of the tangent measure equals \(D\). Since the former coincides with the Hausdorff dimension of \(\Phi\) we obtain an earlier result of \textit{R. D. Mauldin} and \textit{S. C. Williams} [Trans. Am. Math. Soc. 295, 325-346 (1986; Zbl 0625.54047)] as a corollary.
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 569003
zbMATH Keywords: self-similar compact sets
zbMATH Keywords: existence of a fractional tangent measure
zbMATH Keywords: self-similar random measure
zbMATH Keywords: random contraction ratios
zbMATH Keywords: Hausdorff dimension
author: Norbert Patzschke
author: Martina Zähle
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01192964
OpenAlex ID: W2041793176
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 569003