High extremes of Gaussian chaos processes: a discrete time approximation approach

Let $\mathbf{\boldsymbol{\xi}}(t)=(\xi_{1}(t),\ldots,\xi_{d}(t))$ be a Gaussian zero mean stationary a.s. continuous vector process. Let $g\colon{\mathbb{R}}^{d}\to {\mathbb{R}}$ be a homogeneous function of positive degree. We study probabilities of high extrema of the Gaussian chaos process $g(\mathbf{\boldsymbol{\xi}}(t))$. Important examples are products of Gaussian processes, $\prod_{i=1}^{d}\xi_{i}(t)$, and quadratic forms $\sum_{i,j=1}^{d}a_{ij}\xi_{i}(t)\xi_{j}(t)$. Methods of our studies include the Laplace saddle point asymptotic approximation and the double sum asymptotic method for probabilities of high excursions of Gaussian processes. For the first time, using the double sum method, we apply the discrete time approximation with refining grid.