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On the Macroscopic Fractal Geometry of Some Random Sets

  • Davar Khoshnevisan
  • Yimin Xiao
Conference paper
Part of the Progress in Probability book series (PRPR, volume 72)

Abstract

This paper is concerned mainly with the macroscopic fractal behavior of various random sets that arise in modern and classical probability theory. Among other things, it is shown here that the macroscopic behavior of Boolean coverage processes is analogous to the microscopic structure of the Mandelbrot fractal percolation. Other, more technically challenging, results of this paper include:
  1. (i)

    The computation of the macroscopic Minkowski dimension of the graph of a large family of Lévy processes; and

     
  2. (ii)

    The determination of the macroscopic monofractality of the extreme values of symmetric stable processes.

     

As a consequence of (i), it will be shown that the macroscopic fractal dimension of the graph of Brownian motion differs from its microscopic fractal dimension. Thus, there can be no scaling argument that allows one to deduce the macroscopic geometry from the microscopic. Item (ii) extends the recent work of Khoshnevisan et al. (Ann Probab, to appear) on the extreme values of Brownian motion, using a different method.

Keywords

Boolean models Lévy processes Macroscopic Minkowski dimension 

AMS 2010 Subject Classification

Primary 60G51; Secondary 28A80 60G17 60G52 

Notes

Acknowledgements

Research supported in part by the National Science Foundation grant DMS-1307470, DMS-1608575 and DMS-1607089.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  2. 2.Department of Statistics & ProbabilityMichigan State UniversityEast LansingUSA

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