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

Theory Probability and its Applications. 2015. Vol. 60. No. 3. P. 520–527.

A. I. Zhdanov.

Let $(X(t),Y(t))$, $t\ge0$, be a zero-mean stationary Gaussian vector process with a covariance functions for components $r_i(t)$ satisfying Pickand's condition $r_i(t)=1-c_i|t|^{\alpha_i}(1+o(1))$, $t\to 0$, $c_i>0$, $0<\alpha_i\le2$, $i=1,2.$ Let $r_i(t)<1$, $i=1,2$, $t>0.$ Assuming that $r\equiv {\bf E}\,X(t)Y(t)\in(-1,1)$ and $\lim_{t,s\rightarrow0}({\bf E}\,X(t)Y(s)-r)/|t-s|^{\min(\alpha_1,\alpha_2)}$ exists, we study the behavior of probability ${\bf P}(\max_{t\in\lbrack0,p]}X(t)Y(t)>u)$ as $u\rightarrow\infty$ for any $p$. In particular, we derive for any $p$ the exact asymptotic behavior of probability ${\bf P}(\max_{t\in[0,p]}(X^{2}(t)-Y^{2}(t))>u)$ as $u\rightarrow\infty$ for independent Gaussian stationary processes $X(t),Y(t)$ satisfying the above conditions.

Keywords: Гауссовские процессы, Gaussian processes, Gaussian chaos, High extremes probabilities, Double sum method, Гауссовский хаос, Вероятности высоких выбросов, Метод двойных сумм