| Title | Local maxima of Gaussian fields |
| Authors | Georg Lindgren |
| Alternative Location | http://dx.doi.org/10.1007/B..., Restricted Access |
| Publication | Arkiv för Matematik |
| Year | 1972 |
| Volume | 10 |
| Issue | 1-2 |
| Pages | 195 - 218 |
| Document type | Article |
| Status | Published |
| Quality controlled | Yes |
| Language | eng |
| Publisher | Springer Netherlands |
| Abstract English | 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> In Section 1 it is shown that if ξ has a local maximum with heightu at0 then ξ(t) can be expressed as<br> $$\xi _u (t) = uA(t) - \xi _u^\prime b(t) + \Delta (t),$$<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> 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> 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. |
| ISBN/ISSN/Other | ISSN: 1871-2487 online |
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