Probabilities of high extremes for a Gaussian stationary process in a random environment

A. O. Kleban ; M. V. Korulin

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2017), pp. 11-16

Résumé

Let $\xi\left(t\right)$ be a zero-mean stationary Gaussian process with the covariance function $r\left(t\right)$ of Pickands type, i.e., $r(t)=1-|t|^{\alpha}+o(|t|^{\alpha}),~t\to 0,~0\alpha\leq2$, and $\eta\left(t\right), \zeta\left(t\right)$ be periodic random processes. For any $T>0$ and independent $\xi\left(t\right)$, $\eta\left(t\right)$, $\zeta\left(t\right)$ we obtain the exact asymptotic behaviour of the probabilities $P(\max_{t\in[0,T]} \eta\left(t\right) \xi\left(t\right) > u)$, $P(\max_{t\in[0,T]} \left(\xi\left(t\right) + \eta\left(t\right)\right) > u)$ and $P(\max_{t\in[0,T]} \left(\eta\left(t\right) \xi\left(t\right) + \zeta\left(t\right)\right) > u)$ for $u \to \infty$.