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Teor. Veroyatnost. i Primenen., 2001, Volume 46, Issue 1, Pages 94–116 (Mi tvp3953)  

This article is cited in 18 scientific papers (total in 18 papers)

Sample Path Properties of Operator-Slef-Similar Gaussian Random Fields

J. D. Masona, Xiao Yiminb

a University of Utah, Department of Mathematics
b Michigan State University, Department of Statistics and Probability

Abstract: We study the Hausdorff dimension of the image and graph set, hitting probabilities, transience, and other sample path properties of certain isotropic operator-self-similar Gaussian random fields $X = \{X(t), t \in{\mathbf R}^N\}$ with stationary increments, including multiparameter operator fractional Brownian motion. Our results show that if $X({\mathbf 1})$, where ${\mathbf 1}=(1,0,…,0)\in{\mathbf R}^N$, is full, then many of such sample path properties are completely determined by the real parts of the eigenvalues of the self-similarity exponent $D$.

Keywords: operator-self-similar Gaussian random fields, image, graph, Hausdorff dimension, polar set, transience.


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English version:
Theory of Probability and its Applications, 2002, 46:1, 58–78

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Received: 07.04.1999

Citation: J. D. Mason, Xiao Yimin, “Sample Path Properties of Operator-Slef-Similar Gaussian Random Fields”, Teor. Veroyatnost. i Primenen., 46:1 (2001), 94–116; Theory Probab. Appl., 46:1 (2002), 58–78

Citation in format AMSBIB
\by J.~D.~Mason, Xiao~Yimin
\paper Sample Path Properties of Operator-Slef-Similar Gaussian Random Fields
\jour Teor. Veroyatnost. i Primenen.
\yr 2001
\vol 46
\issue 1
\pages 94--116
\jour Theory Probab. Appl.
\yr 2002
\vol 46
\issue 1
\pages 58--78

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    This publication is cited in the following articles:
    1. Xiao Yimin, “The packing measure of the trajectories of multiparameter fractional Brownian motion”, Math. Proc. Cambridge Philos. Soc., 135:2 (2003), 349–373  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. Benson D.A., Meerschaert M.M., Baeumer B., Scheffler H.P., “Aquifer operator scaling and the effect on solute mixing and dispersion”, Water Resour. Res., 42:1 (2006), W01415, 18 pp.  crossref  adsnasa  isi  scopus
    3. Wu DongSheng, Xiao YiMin, “Uniform dimension results for Gaussian random fields”, Sci. China Ser. A, 52:7 (2009), 1478–1496  crossref  mathscinet  zmath  isi  scopus
    4. Biermé H., Lacaux C., “Holder regularity for operator scaling stable random fields”, Stochastic Process. Appl., 119:7 (2009), 2222–2248  crossref  mathscinet  zmath  isi  scopus
    5. Ayache A., Roueff F., Xiao Yimin, “Linear fractional stable sheets: Wavelet expansion and sample path properties”, Stochastic Process. Appl., 119:4 (2009), 1168–1197  crossref  mathscinet  zmath  isi  scopus
    6. Xiao Yimin, “Sample path properties of anisotropic Gaussian random fields”, A minicourse on stochastic partial differential equations, Lecture Notes in Math., 1962, Springer, Berlin, 2009, 145–212  crossref  mathscinet  zmath  isi  scopus
    7. Didier G., Pipiras V., “Integral representations and properties of operator fractional Brownian motions”, Bernoulli, 17:1 (2011), 1–33  crossref  mathscinet  zmath  isi  scopus
    8. Li Yu., Xiao Y., “Multivariate operator-self-similar random fields”, Stochastic Process Appl, 121:6 (2011), 1178–1200  crossref  mathscinet  zmath  isi  scopus
    9. Chen ZhenLong, Xiao YiMin, “On Intersections of Independent Anisotropic Gaussian Random Fields”, Sci. China-Math., 55:11 (2012), 2217–2232  crossref  mathscinet  zmath  isi  scopus
    10. Didier G., Pipiras V., “Exponents, Symmetry Groups and Classification of Operator Fractional Brownian Motions”, J. Theor. Probab., 25:2 (2012), 353–395  crossref  mathscinet  zmath  isi  scopus
    11. Dai H., “Convergence in Law to Operator Fractional Brownian Motions”, J. Theor. Probab., 26:3 (2013), 676–696  crossref  mathscinet  zmath  isi  scopus
    12. Dai H.Sh., “Convergence in Law to Operator Fractional Brownian Motion of Riemann-Liouville Type”, Acta. Math. Sin.-English Ser., 29:4 (2013), 777–788  crossref  mathscinet  zmath  isi  scopus
    13. Bierme H., Lacaux C., “Modulus of Continuity of Some Conditionally Sub-Gaussian Fields, Application To Stable Random Fields”, Bernoulli, 21:3 (2015), 1719–1759  crossref  mathscinet  zmath  isi  scopus
    14. Dai H., Shen G., Kong L., “Limit Theorems For Functionals of Gaussian Vectors”, Front. Math. China, 12:4 (2017), 821–842  crossref  mathscinet  zmath  isi  scopus
    15. Pipiras V. Taqqu M., “Long-Range Dependence and Self-Similarity”, Long-Range Dependence and Self-Similarity, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge Univ Press, 2017, 1–668  crossref  mathscinet  zmath  isi
    16. Soenmez E., “The Hausdorff Dimension of Multivariate Operator-Self-Similar Gaussian Random Fields”, Stoch. Process. Their Appl., 128:2 (2018), 426–444  crossref  mathscinet  zmath  isi  scopus
    17. Dai H., Shen G., Xia L., “Operator Fractional Brownian Sheet and Martingale Differences”, Bull. Korean. Math. Soc., 55:1 (2018), 9–23  crossref  mathscinet  zmath  isi  scopus
    18. Abry P., Didier G., “Wavelet Eigenvalue Regression For N-Variate Operator Fractional Brownian Motion”, J. Multivar. Anal., 168 (2018), 75–104  crossref  mathscinet  zmath  isi  scopus
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