# Lund University Publications

## LUND UNIVERSITY LIBRARIES

### Local maxima of Gaussian fields

(1972) In Arkiv för matematik 10(1-2). p.195-218
Abstract
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 –, 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, , , 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, θu is a... (More)
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 –, 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, , , 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, 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 heightu fort=0; see  for terminology.

In Section 2 we examine the process ξu(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 ξu(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 ξu(t) will have a local minimum at some point τu neart 0, and that τu-t 0 after a normalization is asymptoticallyn-variate normal asu→∞. This is in accordance with the one-dimensional case. (Less)
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Arkiv för matematik
volume
10
issue
1-2
pages
195 - 218
publisher
Springer
external identifiers
• scopus:0000868825
ISSN
0004-2080
DOI
10.1007/BF02384809
language
English
LU publication?
yes
id
edb4ce11-d21f-4016-b0b6-940df36a9fb4 (old id 1273132)
2016-04-01 12:11:19
date last changed
2020-03-08 04:15:39
@article{edb4ce11-d21f-4016-b0b6-940df36a9fb4,
abstract     = {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 –, 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, , , who obtains results of a slightly different type.<br/><br>
In Section 1 it is shown that if ξ has a local maximum with heightu at0 then ξ(t) can be expressed as<br/><br>
$$\xi _u (t) = uA(t) - \xi _u^\prime b(t) + \Delta (t),$$<br/><br>
WhereA(t) andb(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 heightu fort=0; see  for terminology.<br/><br>
In Section 2 we examine the process ξu(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.<br/><br>
In Section 3 we study the locations of the local minima of ξu(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 ξu(t) will have a local minimum at some point τu neart 0, and that τu-t 0 after a normalization is asymptoticallyn-variate normal asu→∞. This is in accordance with the one-dimensional case.},
author       = {Lindgren, Georg},
issn         = {0004-2080},
language     = {eng},
number       = {1-2},
pages        = {195--218},
publisher    = {Springer},
series       = {Arkiv för matematik},
title        = {Local maxima of Gaussian fields},
url          = {http://dx.doi.org/10.1007/BF02384809},
doi          = {10.1007/BF02384809},
volume       = {10},
year         = {1972},
}