Sojourn times and the fragility index

We investigate the sojourn time above a high threshold of a continuous stochastic process Y=(Yt)t∈[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 a max-stable process. This limit coincides with the limit of the fragility index corresponding to (Yi/n)1≤i≤n(Yi/n)1≤i≤n as nn 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. A max-stable process is a prominent example. Given that there is an exceedance at t0t0 above the threshold, we can also compute the asymptotic distribution of the excursion time, which the process spends above the threshold function.