Extremes and Limit Theorems for Difference of Chi-type processes

Abstract

Let $\{{\zeta }_{m,k}^{(\kappa )}(t),t\ge 0\},\kappa >0$ be random processes defined as the differences of two independent stationary chi-type processes with $m$ and $k$ degrees of freedom. In applications such as physical sciences and engineering dealing with structure reliability, of interest is the approximation of the probability that the random process ${\zeta }_{m,k}^{(\kappa )}$ stays in some safety region up to a fixed time $T$. In this paper we derive the asymptotics of $\mathbb{P}\{{\sup }_{t\in [0,T]}{\zeta }_{m,k}^{(\kappa )}(t)>u\},u\to \infty$ under some assumptions on the covariance structures of the underlying Gaussian processes. Further, we establish a Berman sojourn limit theorem and a Gumbel limit result.

To appear in ESAIM P&S