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Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators

Author

Listed:
  • Eduard Belinsky

    (University of the West Indies)

  • Werner Linde

    (Friedrich-Schiller-Universität Jena)

Abstract

We investigate the small ball problem for d-dimensional fractional Brownian sheets by functional analytic methods. For this reason we show that integration operators of Riemann–Liouville and Weyl type are very close in the sense of their approximation properties, i.e., the Kolmogorov and entropy numbers of their difference tend to zero exponentially. This allows us to carry over properties of the Weyl operator to the Riemann–Liouville one, leading to sharp small ball estimates for some fractional Brownian sheets. In particular, we extend Talagrand's estimate for the 2-dimensional Brownian sheet to the fractional case. When passing from dimension 1 to dimension d≥2, we use a quite general estimate for the Kolmogorov numbers of the tensor products of linear operators.

Suggested Citation

  • Eduard Belinsky & Werner Linde, 2002. "Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators," Journal of Theoretical Probability, Springer, vol. 15(3), pages 589-612, July.
    • Handle: RePEc:spr:jotpro:v:15:y:2002:i:3:d:10.1023_a:1016263614257
    DOI: 10.1023/A:1016263614257
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    References listed on IDEAS

    as
    1. David M. Mason & Zhan Shi, 2001. "Small Deviations for Some Multi-Parameter Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 14(1), pages 213-239, January.
    2. Thomas Dunker, 2000. "Estimates for the Small Ball Probabilities of the Fractional Brownian Sheet," Journal of Theoretical Probability, Springer, vol. 13(2), pages 357-382, April.
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    Cited by:

    1. Fuchang Gao & Wenbo V. Li, 2006. "Logarithmic Level Comparison for Small Deviation Probabilities," Journal of Theoretical Probability, Springer, vol. 19(3), pages 535-556, December.
    2. Alexandre Richard, 2017. "Some Singular Sample Path Properties of a Multiparameter Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1285-1309, December.
    3. Fuchang Gao & Wenbo V. Li, 2007. "Logarithmic Level Comparison for Small Deviation Probabilities," Journal of Theoretical Probability, Springer, vol. 20(1), pages 1-23, March.
    4. Helga Schack, 2009. "An Optimal Wavelet Series Expansion of the Riemann–Liouville Process," Journal of Theoretical Probability, Springer, vol. 22(4), pages 1030-1057, December.

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