The exact Hausdorff measure for random re-ordering of Cantor set
type: Item
instance of: scholarly article
title: The exact Hausdorff measure for random re-ordering of Cantor set
zbMATH Open document ID: 0834.28003
author: Xiaoyu Hu
published in: Science in China. Series A
publication date: 1996-04-23T00:00:00Z
review text: Let \(X_1, X_2,\ldots\) be a sequence of i.i.d. random variables uniformly distributed on \([0,1]\), and let \(Q_n\), \(n \geq 1\), be random sets of integers defined by \(Q_n = \{m \geq 1 : X_m < X_n\}\). For a sequence of reals \(\{a_n\}^\infty_{n = 1}\) with \(\sum a_n = 1\) set \(t_n = \sum_{s \in Q_n} a_s\) and \(J_n = (t_n, t_n + a_n)\). The author finds the packing dimension for the random set \(K = [0,1] \backslash \bigcup^\infty_{n = 1} J_n\). In particular, if \(a_n = 3^{-n}\), \(n \geq 1\), then \(K\) is a random Cantor set obtained by random reordering of complementary intervals to the classical Cantor set. In this case the author derives the exact Hausdorff and packing measures for \(K\).
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 783472
zbMATH Keywords: random fractal
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: packing measure
zbMATH Keywords: packing dimension
zbMATH Keywords: random Cantor set
reviewed by: Ilya S. Molchanov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 783472