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A series expansion of fractional Brownian motion
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  • Published: 03 March 2004

A series expansion of fractional Brownian motion

  • Kacha Dzhaparidze1 &
  • Harry van Zanten2 

Probability Theory and Related Fields volume 130, pages 39–55 (2004)Cite this article

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  • 106 Citations

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Abstract.

Let B be a fractional Brownian motion with Hurst index H∈(0,1). Denote by the positive, real zeros of the Bessel function J −H of the first kind of order −H, and let be the positive zeros of J 1−H . In this paper we prove the series representation where X 1 ,X 2 ,... and Y 1 ,Y 2 ,... are independent, Gaussian random variables with mean zero and and the constant c H 2 is defined by c H 2=π−1Γ(1+2H) sin πH. We show that with probability 1, both random series converge absolutely and uniformly in t∈[0,1], and we investigate the rate of convergence.

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Author information

Authors and Affiliations

  1. Center for Mathematics and Computer Science Kruislaan 413, 1098 SJ Amsterdam, The Netherlands

    Kacha Dzhaparidze

  2. Faculty of Sciences, Vrije Universiteit Amsterdam Department of Mathematics, De Boelelaan 1081a, 1081, HV Amsterdam, The Netherlands

    Harry van Zanten

Authors
  1. Kacha Dzhaparidze
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  2. Harry van Zanten
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Corresponding author

Correspondence to Kacha Dzhaparidze.

Additional information

Mathematics Subject Classification (2000): 60G15, 60G18, 33C10

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Cite this article

Dzhaparidze, K., Zanten, H. A series expansion of fractional Brownian motion. Probab. Theory Relat. Fields 130, 39–55 (2004). https://doi.org/10.1007/s00440-003-0310-2

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  • Received: 23 September 2002

  • Revised: 27 September 2003

  • Published: 03 March 2004

  • Issue Date: September 2004

  • DOI: https://doi.org/10.1007/s00440-003-0310-2

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Keywords

  • Fractional Brownian motion
  • Series expansion
  • Bessel function
  • Hankel transform
  • Fractional calculus
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