A representation for self-similar processes

Murad S. Taqqu

Stochastic Processes and their Applications, 1978, vol. 7, issue 1, 55-64

Abstract: A self-similar process Z(t) has stationary increments and is invariant in law under the transformation Z(i)-->c-HZ(ct), c[greater-or-equal, slanted]0. The choice 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 , m=1,2,..., whose finite-dimensional moments have been specified in an earlier paper. is the Gaussian fractional Brownian motion but the process are not Gaussian when m[greater-or-equal, slanted]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".

Date: 1978
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