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A packing dimension theorem for Gaussian random fields

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  • Xiao, Yimin

Abstract

Let be a Gaussian random field with values in defined by where X1,...,Xd are independent copies of a centered Gaussian random field X0. Under certain general conditions, Xiao [Xiao, Y., 2007. Strong local nondeterminism and the sample path properties of Gaussian random fields. In: Lai, Tze Leung, Shao, Qiman, Qian, Lianfen (Eds.), Asymptotic Theory in Probability and Statistics with Applications. Higher Education Press, Beijing, pp. 136-176] defined an upper index [alpha]* and a lower index [alpha]* for X0 and showed that the Hausdorff dimensions of the range X([0,1]N) and graph are determined by the upper index [alpha]*. In this paper, we prove that the packing dimensions of X([0,1]N) and are determined by the lower index [alpha]* of X0. Namely, and This verifies a conjecture of Xiao in the above-cited reference. Our method is based on the potential-theoretic approach to packing dimension due to Falconer and Howroyd [Falconer, K.J., Howroyd, J.D., 1997. Packing dimensions for projections and dimension profiles. Math. Proc. Cambridge Philos. Soc. 121, 269-286].

Suggested Citation

  • Xiao, Yimin, 2009. "A packing dimension theorem for Gaussian random fields," Statistics & Probability Letters, Elsevier, vol. 79(1), pages 88-97, January.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:1:p:88-97
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    References listed on IDEAS

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    1. Xiao, Yimin, 1997. "Packing dimension of the image of fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 33(4), pages 379-387, May.
    2. Jean‐Marc Bardet & Pierre Bertrand, 2007. "Identification of the multiscale fractional Brownian motion with biomechanical applications," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(1), pages 1-52, January.
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    Cited by:

    1. Li, Jinjun, 2011. "A class of probability distribution functions preserving the packing dimension," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1782-1791.
    2. Du, Yali & Miao, Junjie & Wu, Dongsheng & Xiao, Yimin, 2015. "Packing dimensions of the images of Gaussian random fields," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 209-217.

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