A representation for self-similar processes

Abstract

A self-similar process Z( t) has stationary increments and is invariant in law under the transformation Z( i)→ c - H Z( ct), c⩾0. The choice 1 2 <H<1 ensures that the increments of Z( t) exhibit a long range positive correlation. Mandelbrot and Van Ness investigated the case where Z( t) is Gaussian and represented that Gaussian self-similar process as a fractional integral of Brownian motion. They called it fractional Brownian motion. This paper provides a time-indexed representation for a sequence of self- similar processes Z ̄ m(t) , m=1,2,…, whose finite-dimensional moments have been specified in an earlier paper. Z ̄ 1(t) is the Gaussian fractional Brownian motion but the process Z ̄ m(t) are not Gaussian when m⩾2. Self-similar processes are being studied in physics, in the context of the renormalization group theory for critical phenomena, and in hydrology where they account for the so-called “Hurst effect”.