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

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 $σ(s,t)$. Denote by $\mathcal{L}=\{(s,t): σ(s,t)=\max_{(s',t')\in E}σ(s',t')\}$. 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.

No reviews yet.

No comments yet.