On the Macroscopic Fractal Geometry of Some Random Sets

TL;DR

This paper investigates the large-scale fractal properties of various random sets in probability theory, revealing differences between macroscopic and microscopic fractal dimensions and extending previous work on stable processes.

Contribution

It introduces new methods to compute macroscopic fractal dimensions of complex stochastic processes and compares macroscopic and microscopic fractal behaviors.

Findings

Macroscopic fractal dimension of Brownian graph differs from microscopic dimension

Boolean coverage processes resemble Mandelbrot fractal percolation

Extended analysis of extreme values of symmetric stable processes

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: (i) The computation of the macroscopic dimension of the graph of a large family of L\'evy processes; and (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, Kim, and Xiao \cite{KKX} on the extreme values of Brownian motion, using a different method.