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Validity of the expected Euler characteristic heuristic. (English) Zbl 1083.60031
Authors’ abstract: We study the accuracy of the expexted Euler characteristic approximation to the distribution of the maximum of a smooth, centered, unit variance Gaussian process \(f\). Using a point process representation of the error, valid for arbitrary smooth processes, we show that the error is in general exponentially smaller than any of the terms in the approximation. We also give a lower bound on this exponential rate of decay in terms of the maximal variance of a family of Gaussian processes \(f^x\), derived from the original process \(f\).

60G15 Gaussian processes
60G60 Random fields
53A17 Differential geometric aspects in kinematics
58A05 Differentiable manifolds, foundations
60G17 Sample path properties
62M40 Random fields; image analysis
60G70 Extreme value theory; extremal stochastic processes
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