The Weierstrass Function and Fractional Brownian Motion
Murad Taqqu
Department of Mathematics and Statistics
Boston University
Abstract
Benoit Mandelbrot suggested that Weierstrass' nowhere differentiable
function can be modified and randomized so as to approximate
fractional Brownian motion, which is a Gaussian self-similar process
whose paths are almost surely non-differentiable. The randomization
involves introducing independent and identically distributed random
variables with finite variance in the definition of the Weierstrass
function. We will show how one then obtains fractional Brownian motion
in the limit.
If time allows we will describe various modifications and indicate
what happens if, for example, one introduces in the above
randomization strongly dependent random variables instead of
independent ones or if one uses infinite variance random variables
instead of finite variance ones.
This is joint work with Vladas Pipiras.