Sojourn times of Gaussian related random fields

TL;DR

This paper explores the asymptotic behavior of sojourn times in Gaussian-related random fields, revealing a reciprocal relationship with supremum tail asymptotics and introducing a new uniform double-sum method.

Contribution

It establishes a general framework linking tail asymptotics of sojourn times and supremum, and demonstrates how supremum asymptotics can be used to derive sojourn time behavior in Gaussian fields.

Findings

Established a relationship between tail asymptotics of sojourn times and supremum.

Introduced the uniform double-sum method for tail asymptotics.

Applied findings to Gaussian fields, chi-processes, and queueing processes.

Abstract

This paper is concerned with the asymptotic analysis of sojourn times of random fields with continuous sample paths. Under a very general framework we show that there is an interesting relationship between tail asymptotics of sojourn times and that of supremum. Moreover, we establish the uniform double-sum method to derive the tail asymptotics of sojourn times. In the literature, based on the pioneering research of S. Berman the sojourn times have been utilised to derive the tail asymptotics of supremum of Gaussian processes. In this paper we show that the opposite direction is even more fruitful, namely knowing the asymptotics of supremum o f random processes and fields (in particular Gaussian) it is possible to establish the asymptotics of their sojourn times. We illustrate our findings considering i) two dimensional Gaussian random fields, ii) chi-process generated by stationary Gaussian processes and iii) stationary Gaussian queueing processes.