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A representation for self-similar processes

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  • Taqqu, Murad S.

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".

Suggested Citation

  • Taqqu, Murad S., 1978. "A representation for self-similar processes," Stochastic Processes and their Applications, Elsevier, vol. 7(1), pages 55-64, March.
    • Handle: RePEc:eee:spapps:v:7:y:1978:i:1:p:55-64
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    Cited by:

    1. Chi, Zhiyi, 2001. "Stationary self-similar random fields on the integer lattice," Stochastic Processes and their Applications, Elsevier, vol. 91(1), pages 99-113, January.
    2. Christian Mönch, 2022. "Universality for Persistence Exponents of Local Times of Self-Similar Processes with Stationary Increments," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1842-1862, September.
    3. Bardet, Jean-Marc & Tudor, Ciprian, 2014. "Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt process," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 1-16.
    4. Bardet, J.-M. & Tudor, C.A., 2010. "A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2331-2362, December.

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