Extremes and Limit Theorems for Difference of Chi-type processes

TL;DR

This paper derives asymptotic probabilities and limit theorems for the supremum of difference processes of chi-type processes, relevant for reliability analysis in engineering and physical sciences.

Contribution

It introduces new asymptotic results and limit theorems for the supremum of difference chi-type processes, extending understanding of their extreme behavior.

Findings

Asymptotic approximation of tail probabilities for supremum of difference chi-type processes.

Berman sojourn limit theorem established for these processes.

Gumbel limit law derived for the maximum distribution.

Abstract

Let $\{\zeta_{m,k}^{(\kappa)}(t), t \ge0\}, \kappa>0$ 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.