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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Asymptotic properties of Gaussian random fields


Authors: Clifford Qualls and Hisao Watanabe
Journal: Trans. Amer. Math. Soc. 177 (1973), 155-171
MSC: Primary 60G15
DOI: https://doi.org/10.1090/S0002-9947-1973-0322943-8
MathSciNet review: 0322943
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Abstract: In this paper we study continuous mean zero Gaussian random fields $ X(p)$ with an N-dimensional parameter and having a correlation function $ \rho (p,q)$ for which $ 1 - \rho (p,q)$ is asymptotic to a regularly varying (at zero) function of the distance $ {\text{dis}}\;(p,q)$ with exponent $ 0 < \alpha \leq 2$. For such random fields, we obtain the asymptotic tail distribution of the maximum of $ X(p)$ and an asymptotic almost sure property for $ X(p)$ as $ \vert p\vert \to \infty $. Both results generalize ones previously given by the authors for $ N = 1$.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0322943-8
Keywords: Regular variation, slow variation, random fields, supremum of stochastic processes, isotropic, stationary, 0-1 law
Article copyright: © Copyright 1973 American Mathematical Society