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Average densities of the image and zero set of stable processes

Author

Listed:
  • Falconer, K. J.
  • Xiao, Y. M.

Abstract

The 'order-two' or 'average' density of a measure [mu] at a point x is defined as limT --> x(1/T)[integral operator]T0[mu](B(x, e-s))e[alpha]sds for appropriate [alpha]. We show that, with probability one, the order-two density of the natural measure [mu] on the image set or zero set of a wide class of stable processes exists and takes the same value almost everywhere in the support of [mu]. We calculate this value in certain cases.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:55:y:1995:i:2:p:271-283
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    References listed on IDEAS

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    1. Patzschke, N. & Zähle, M., 1993. "Fractional differentiation in the self-affine case II - Extremal processes," Stochastic Processes and their Applications, Elsevier, vol. 45(1), pages 61-72, March.
    2. 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.
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    Cited by:

    1. 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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    1. 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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