Titlebook: Random Fields and Geometry; Robert J. Adler,Jonathan E. Taylor Book 2007 Springer-Verlag New York 2007 Area.Gaussian process.Volume.astrop

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Gaussian Inequalities comparison to the power and simplicity of the coresponding basic inequality of Gaussian processes. This inequality was discovered independently, and established with very different proofs, by Borell [30] and Tsirelson, Ibragimov, and Sudakov (TIS) [160]. For brevity, we shall call it the Borell–TIS

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Excursion Probabilitiesto evaluate the . . where . is a random process over some parameter set . . As usual, we shall restrict ourselves to the case in which . is centered and Gaussian and . is compact for the canonical metric of (1.3.1).

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insurrection

Differential Geometrythe format of a “glossary of terms. ” Most will be familiar to those who have taken a couple courses in differential geometry, and hopefully informative enough to allow the uninitiated. to follow the calculations in later chapters. However, to go beyond merely following the arguments there and to re

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Critical Point Theorynifolds of one kind or another, which will serve there as parameter spaces for our random fields, as well as appearing in the proofs. The second are the Lipschitz–Killing curvatures that we met briefly in Chapter 7 and shall look at far more closely, in the piecewise smooth scenario, in Chapter 10.

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Volume of Tubes to begin to reap the benefits of our investment, while at the same time developing some themes a little further for later exploitation. This chapter focuses on the celebrated volume-of-tubes formula of Wey1 [73, 168], which expresses the Lebesgue volume of a tube of radius ρ around a set . embedded

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