The Geometry of Random Fields

A₁ bounded Brownian motion Brownian sheet Chapter compact components condition of order consider continuous convergence coordinate covariance function cube defined denote density differentiable distribution function dp(u ergodic erraticism example excursion characteristics excursion sets exists finite Firstly following lemma following result following theorem Furthermore Gaussian field Gaussian process Gaussian random field given Hausdorff dimension Hölder condition homogeneous Gaussian IG characteristic implies inequality integral geometry interval isotropic Lebesgue measure Let X(t level crossings matrix maxima mean number mean square mean value non-negative notation number of points obtain one-dimensional parameter partial derivatives probability properties random field X(t random variable real-valued sample functions sample paths satisfying second-order sequence splitting field stochastic subset suitable regularity t₁ theory variance vector write X₁ X₁(t x² field zero zero-mean