Local maxima of Gaussian fields
Lindgren, Georg LU
(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 [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, 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 [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, 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 [4] 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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1273132
- author
- Lindgren, Georg
LU
- organization
- publishing date
- 1972
- 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)
- date added to LUP
- 2016-04-01 12:11:19
- date last changed
- 2021-08-22 05:20:31
@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 [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.<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 [4] 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}},
}