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1972 Local maxima of Gaussian fields
Georg Lindgren
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Ark. Mat. 10(1-2): 195-218 (1972). DOI: 10.1007/BF02384809


The structure of a stationary Gaussian process near a local maximum with a prescribed height u has been explored in several papers by the present author, see [5]–[7], which include results for moderate u as well as for u→±∞. In this paper we generalize these results to a homogeneous Gaussian field {ξ(t) t ∈ Rn}, with mean zero and the covariance function r(t). The local structure of a Gaussian field near a high maximum has also been studied by Nosko, [8], [9], who obtains results of a slightly different type.

In Section 1 it is shown that if ξ has a local maximum with height u at 0 then ξ(t) can be expressed as $\xi _u (t) = uA(t) - \xi _u^\prime b(t) + \Delta (t),$

WhereA(t) and b(t) are certain functions, θu is a random vector, and Δ(t) is a non-homogeneous Gaussian field. Actually ξu(t) is the old process ξ(t) conditioned in the horizontal window sense to have a local maximum with height u for t=0; see [4] for terminology.

In Section 2 we examine the process ξu(t) as u→−∞, and show that, after suitable normalizations, it tends to a fourth degree polynomial in t1…, tn with random coefficients. This result is quite analogous with the one-dimensional case.

In Section 3 we study the locations of the local minima of ξu(t) as u → ∞. In the non-isotropic case r(t) may have a local minimum at some point t0. Then it is shown in 3.2 that ξu(t) will have a local minimum at some point τu near t0, and that τu-t0 after a normalization is asymptotically n-variate normal as u→∞. This is in accordance with the one-dimensional case.

Funding Statement

This research was supported in part by the Office of Naval Research under Contract N00014-67-A-0002.


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Georg Lindgren. "Local maxima of Gaussian fields." Ark. Mat. 10 (1-2) 195 - 218, 1972.


Received: 22 November 1971; Published: 1972
First available in Project Euclid: 31 January 2017

zbMATH: 0251.60031
MathSciNet: MR319259
Digital Object Identifier: 10.1007/BF02384809

Rights: 1972 © Institut Mittag-Leffler

Vol.10 • No. 1-2 • 1972
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