Local maxima of Gaussian fields

Abstract

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 {Mathematical expression} Where A(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. © 1972 Institut Mittag-Leffler.