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Model fields in crossing theory: A weak convergence perspective. (English) Zbl 0663.60041

At the beginning of this article, the author gives a brief review of the literature concerning model fields and weak convergence of probability measures on spaces of smooth functions. Then, following the author and R. J. Adler [ibid. 14, 543-565 (1982; Zbl 0487.60044)], he constructs such a model field \(X_ u\) on a regular Gaussian random field X(t), with index \(t\in {\mathbb{R}}^ n\) and smooth covariance function, using the horizontal window conditioning method and a weak convergence result on \((C^ 2,{\mathcal C}^ 2)\), where \(C^ 2\) is the usual space of functions with continuous derivatives up to order two and \({\mathcal C}^ 2\) a \(\sigma\)-field generated by a particular metric.
Then the author shows that, if X is ergodic, the distribution of \(X_ u\), in some subspace of \((C^ 2,{\mathcal C}^ 2)\), is the weak limit as \(r\to \infty\) of the empirical distribution of X around the locations in the closed ball with center 0 and radius r of upcrossings of u in the n th direction. He also proves that the distribution of \(X_ u(t)\), in \((C^ 2,{\mathcal C}^ 2)\), is the same as that of X when \(| t| \to \infty\), i.e. at large distance from upcrossings of u in the n th direction at \(t=0.\)
The article concludes with the full weak convergence, when \(u\to \infty\), of a normalization of \(X_ u\) to a random elliptical paraboloid, and with the study of the Lebesgue measure of a particular excursion set of \(X_ u\) jointly with the height of this set for large u.
Reviewer: Ph.Nobelis

MSC:

60G60 Random fields
60G15 Gaussian processes
60F05 Central limit and other weak theorems

Citations:

Zbl 0487.60044
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