Extremes of Gaussian Random Fields with maximum variance attained over smooth curves

Abstract

Let $X(s,t),(s,t)\in E$, with $E\subset {\mathbb{R}}^2$ a compact set, be a centered two dimensional Gaussian random field with continuous trajectories and variance function $\sigma (s,t)$. Denote by $\mathcal{L}=\{(s,t):\sigma (s,t)={\max }_{(s^{\prime },t^{\prime })\in E}\sigma (s^{\prime },t^{\prime })\}$. In this contribution, we derive the exact asymptotics of $\mathbb{P}\left({\sup }_{(s,t)\in E}X(s,t)>u\right)$, as $u\to \infty$, under condition that $\mathcal{L}$ is a smooth curve. We illustrate our findings by an application concerned with extremes of the aggregation of two independent fractional Brownian motions.

26 pages

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