The Geometry of Random Fields
Originally published in 1981, The Geometry of Random Fields remains an important text for its coverage and exposition of the theory of both smooth and nonsmooth random fields; closed form expressions for the various geometric characteristics of the excursion sets of smooth, stationary, Gaussian random fields over N-dimensional rectangles; descriptions of the local behavior of random fields in the neighborhoods of high maxima; and a treatment of the Markov property for Gaussian fields. Audience: researchers in probability and statistics, with no prior knowledge of geometry required. Since the book was originally published it has become a standard reference in areas of physical oceanography, cosmology, and neuroimaging. It is written at a level accessible to nonspecialists, including advanced undergraduates and early graduate students.
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Adler assume behaviour Belyaev Borel bounded Brownian motion Brownian sheet Chapter compact components consider continuous convergence coordinate covariance function cube deﬁned deﬁnition denote density dimensional distribution function DT characteristic equation ergodic erraticism example excursion characteristics excursion sets exists expression fact ﬁnd ﬁnite ﬁrst ﬁrstly ﬁxed following lemma following result following theorem Furthermore Gaussian ﬁeld Gaussian process Gaussian random ﬁeld geometry given Hausdorff dimension Holder condition homogeneous Gaussian implies independent inequality inﬁnite integral geometry interval isotropic Lebesgue measure Lemma level crossings matrix maxima mean number mean square mean value negative deﬁnite non-negative notation number of points number of upcrossings obtain ofthe one-dimensional parameter partial derivatives probability problem properties prove random ﬁelds random variable real-valued representation sample functions sample paths satisﬁes satisfying second-order sequence speciﬁc stochastic stochastic process subset suitable regularity theory variance vector write yields zero zero-mean