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Small Deviations for Some Multi-Parameter Gaussian Processes

Author

Listed:
  • David M. Mason

    (University of Delaware)

  • Zhan Shi

    (Université Paris VI)

Abstract

We prove some general lower bounds for the probability that a multi-parameter Gaussian process has very small values. These results, when applied to a certain class of fractional Brownian sheets, yield the exact rate for their so-called small ball probability. We show by example how to use such results to compute the Hausdorff dimension of some exceptional sets determined by maximal increments.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:14:y:2001:i:1:d:10.1023_a:1007833401562
    DOI: 10.1023/A:1007833401562
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    References listed on IDEAS

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    1. 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. S. Dereich & F. Fehringer & A. Matoussi & M. Scheutzow, 2003. "On the Link Between Small Ball Probabilities and the Quantization Problem for Gaussian Measures on Banach Spaces," Journal of Theoretical Probability, Springer, vol. 16(1), pages 249-265, January.
    2. 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.
    3. 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.

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    1. 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.
    2. S. Dereich & F. Fehringer & A. Matoussi & M. Scheutzow, 2003. "On the Link Between Small Ball Probabilities and the Quantization Problem for Gaussian Measures on Banach Spaces," Journal of Theoretical Probability, Springer, vol. 16(1), pages 249-265, January.

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