Extremes of homogeneous Gaussian random fields

Let {X (s, t): s, t >= 0} be a centred homogeneous Gaussian field with almost surely continuous sample paths and correlation function r (s, t) = cov(X(s, t), X(0, 0)) such that r(s, t) = 1 - vertical bar s vertical bar(alpha 1) - vertical bar t vertical bar(alpha 2) + o(vertical bar s vertical bar(alpha 1) + vertical bar t vertical bar(alpha 2)), s,t -> 0, with alpha 1, alpha 2 is an element of(0,2], and r (s, t) < 1 for (s, t) not equal (0, 0). In this contribution we derive an asymptotic expansion (as u -> infinity) of P(sup((sn1(u),tn2(u))is an element of[0,x]x[0,y]) X(s,t) <= u), where n(1)(u)n(2)(u) = u(2/alpha 1+2/alpha 2) Psi(u), which holds uniformly for (x, y) is an element of [A, B](2) with A, B two positive constants and Psi the survival function of an N(0, 1) random variable. We apply our findings to the analysis of extremes of homogeneous Gaussian fields over more complex parameter sets and a ball of random radius. Additionally, we determine the extremal index of the discretised random field determined by X (s, t)

Extrema of locally stationary Gaussian fields on growing manifolds

We consider a class of non-homogeneous, continuous, centered Gaussian random fields $\{X_h(t), t \in {\cal M}_h;\,0 < h \le 1\}$ where ${\cal M}_h$ denotes a rescaled smooth manifold, i.e. ${\cal M}_h = \frac{1}{h} {\cal M},$ and study the limit behavior of the extreme values of these Gaussian random fields when $h$ tends to zero, which means that the manifold is growing. Our main result can be thought of as a generalization of a classical result of Bickel and Rosenblatt (1973a), and also of results by Mikhaleva and Piterbarg (1997).Comment: 28 pages, 1 figur

Extremes of the standardized Gaussian noise

Let $\{\xi_n, n\in\Z^d\}$ be a $d$-dimensional array of i.i.d. Gaussian random variables and define \SSS(A)=\sum_{n\in A} \xi_n, where $A$ is a finite subset of $\Z^d$. We prove that the appropriately normalized maximum of \SSS(A)/\sqrt{|A|}, where $A$ ranges over all discrete cubes or rectangles contained in $\{1,\ldots,n\}^d$, converges in the weak sense to the Gumbel extreme-value distribution as $n\to\infty$. We also prove continuous-time counterparts of these results.Comment: 18 page

Extremes of the two-dimensional Gaussian free field with scale-dependent variance

In this paper, we study a random field constructed from the two-dimensional Gaussian free field (GFF) by modifying the variance along the scales in the neighborhood of each point. The construction can be seen as a local martingale transform and is akin to the time-inhomogeneous branching random walk. In the case where the variance takes finitely many values, we compute the first order of the maximum and the log-number of high points. These quantities were obtained by Bolthausen, Deuschel and Giacomin (2001) and Daviaud (2006) when the variance is constant on all scales. The proof relies on a truncated second moment method proposed by Kistler (2015), which streamlines the proof of the previous results. We also discuss possible extensions of the construction to the continuous GFF.Comment: 30 pages, 4 figures. The argument in Lemma 3.1 and 3.4 was revised. Lemma A.4, A.5 and A.6 were added for this reason. Other typos were corrected throughout the article. The proof of Lemma A.1 and A.3 was simplifie