On the Macroscopic Fractal Geometry of Some Random Sets

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.