Local maxima of Gaussian fields

Summary

The structure of a stationary Gaussian process near a local maximum with a prescribed heightu has been explored in several papers by the present author, see [5]–[7], which include results for moderateu as well as foru→±∞. In this paper we generalize these results to a homogeneous Gaussian field {ξ(t) t ∈ R ⁿ}, with mean zero and the covariance functionr(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 heightu at0 then ξ(t) can be expressed as

$$\xi _u (t) = uA(t) - \xi _u^\prime b(t) + \Delta (t),$$

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

In Section 2 we examine the process ξ_u(t) asu→−∞, and show that, after suitable normalizations, it tends to a fourth degree polynomial int ₁…,t _n 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 ξ_u(t) asu → ∞. In the non-isotropic caser(t) may have a local minimum at some pointt ⁰. Then it is shown in 3.2 that ξ_u(t) will have a local minimum at some point τ^u neart ⁰, and that τ^u-t ⁰ after a normalization is asymptoticallyn-variate normal asu→∞. This is in accordance with the one-dimensional case.