High extrema of Gaussian chaos processes

Abstract

Let ξ ( t)=(ξ ₁(t),…,ξ _d (t)) be a Gaussian stationary vector process. Let $g:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}$ be a homogeneous function. We study probabilities of large extrema of the Gaussian chaos process g(ξ(t)). Important examples include $g(\mathbf {\boldsymbol {\xi }}(t))={\prod }_{i=1}^{d}\xi _{i}(t)$ and $g(\mathbf {\boldsymbol {\xi }}(t))={\sum }_{i=1}^{d}a_{i}{\xi _{i}^{2}}(t)$. We review existing results partially obtained in collaboration with E. Hashorva, D. Korshunov, and A. Zhdanov. We also present the principal methods of our investigations which are the Laplace asymptotic method and other asymptotic methods for probabilities of high excursions of Gaussian vector process’ trajectories.