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Fractional differentiation in the self-affine case I - Random functions

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  • Patzschke, N.
  • Zähle, M.

Abstract

The invariance structure of self-affine functions and measures leads to the concept of fractional Cesáro derivatives and densities, respectively. In the present paper the case of random functions from p into q is considered. It is shown that the corresponding derivatives exist a.s. and equal a constant in the ergodic case. Part II will deal with the class of self-similar extremal processes and certain extensions. In Part III the fractional density of the Cantor measure will be evaluated, and arbitrary self-similar random measures will be treated in Part IV. There exists a deeper connection to fractional differentiation in the theory of function spaces which will be established elsewhere.

Suggested Citation

  • Patzschke, N. & Zähle, M., 1992. "Fractional differentiation in the self-affine case I - Random functions," Stochastic Processes and their Applications, Elsevier, vol. 43(1), pages 165-175, November.
    • Handle: RePEc:eee:spapps:v:43:y:1992:i:1:p:165-175
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    Cited by:

    1. Falconer, K. J. & Xiao, Y. M., 1995. "Average densities of the image and zero set of stable processes," Stochastic Processes and their Applications, Elsevier, vol. 55(2), pages 271-283, February.
    2. Mörters, Peter, 1998. "The average density of the path of planar Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 74(1), pages 133-149, May.

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