On probability of high extremes for product of two independent Gaussian stationary processes

Let X(t), Y(t), t≥0, be two independent zero-mean stationary Gaussian processes, whose covariance functions are such that ri(t)=1−|t|ai+o(|t|ai) as t→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\rightarrow 0$\end{document}, with 0<ai≤2, i=1,2 and both of the functions are less than one for non-zero t. We derive for any p the exact asymptotic behavior of the probability P(maxt∈[0,p]X(t)Y(t)>u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P(\max _{t\in \lbrack 0,p]}X(t)Y(t)>u)$\end{document} as u→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u\rightarrow \infty $\end{document}. We discuss possibilities generalizing obtained results to other Gaussian chaos processes h(X(t)), with a Gaussian vector process X(t) and a homogeneous function h of positive order.