Sojourn Times and the Fragility Index

TL;DR

This paper studies the asymptotic behavior of the time a stochastic process spends above high thresholds, linking it to extreme value theory and the fragility index, with applications to generalized Pareto processes.

Contribution

It establishes the existence of the limit of expected sojourn time for processes in the domain of attraction of an extreme value process and provides methods to compute asymptotic distributions.

Findings

Limit of expected sojourn time exists under certain conditions.

Asymptotic distribution of sojourn times can be computed for generalized Pareto processes.

Connection established between sojourn times, fragility index, and extreme value processes.

Abstract

We investigate the sojourn time above a high threshold of a continuous stochastic process Y on [0,1]. It turns out that the limit, as the threshold increases, of the expected sojourn time given that it is positive, exists if the copula process corresponding to Y is in the functional domain of attraction of of an extreme value process. This limit coincides with the limit of the fragility index corresponding to finite (n-)dimensional distributions of Y as n and the threshold increase. If the process is in a certain neighborhood of a generalized Pareto process, then we can replace the constant threshold by a general threshold function and we can compute the asymptotic sojourn time distribution. An extreme value process is a prominent example. Given that there is an exceedance at some t_0 above the threshold, we can also compute the asymptotic distribution of the time cluster length, which the process spends above the threshold function.