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2018 Random Cutouts of the Unit Cube with I.U.D Centers
E. M. Dong, Z. Y. Zhu
Real Anal. Exchange 43(1): 205-220 (2018). DOI: 10.14321/realanalexch.43.1.0205

Abstract

Consider the random open balls \(B_n(\omega):=B(\omega_n,r_n)\) with their centers \(\omega_n\) being i.u.d. on the \(d\)-dimensional unit cube \([0,1]^d\) and with their radii \(r_n\sim cn^{-\frac{1}{d}}\) for some constant \(0<c<(\beta(d))^{-\frac{1}{d}}\), where \(\beta(d)\) is the volume of the \(d\) dimensional unit ball. We call \([0,1]^d-\bigcup_{n=1}^{\infty} B_n(\omega)\) a random cutout set. In this paper, we present an exposition of the Zähle cutout model by a detailed study of such a random cutout set for the purpose of teaching and learning. We show that with probability one the Hausdorff dimension of such random cut-out set is at most \(d(1-\beta(d)c^d)\) and frequently equals \(d(1-\beta(d)c^d)\).

Citation

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E. M. Dong. Z. Y. Zhu. "Random Cutouts of the Unit Cube with I.U.D Centers." Real Anal. Exchange 43 (1) 205 - 220, 2018. https://doi.org/10.14321/realanalexch.43.1.0205

Information

Published: 2018
First available in Project Euclid: 2 May 2018

zbMATH: 06924881
MathSciNet: MR3816439
Digital Object Identifier: 10.14321/realanalexch.43.1.0205

Subjects:
Primary: 26A04 , 28A80
Secondary: 60G57

Keywords: Fractal , martingale , random cut-out set , Random fractal , random measure

Rights: Copyright © 2018 Michigan State University Press

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Vol.43 • No. 1 • 2018
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