On the tail asymptotics of supremum of stationary $\chi$-processes with random trend

Abstract

Let $\chi_n(t)$, $t\geqslant0$, be a chi-process with $n$ degrees of freedom. We derive the asymptotic exact result for$$\mathbf{P}\left(\sup_{t\in [0,T]} (\chi_n(t)+\eta(t))>u\right),\text{ as }u\to\infty,$$where $\eta(t)$ is a certain random process independent of $\chi_n(t)$ and $T>0$ is a constant.