Sojourn times and the fragility index

We investigate the sojourn time above a high threshold of a continuous stochastic process Y = ( Y t ) 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 ( Y i / n ) 1 ≤ i ≤ n as n and the threshold increase. 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 t 0 above the threshold, we can also compute the asymptotic distribution of the excursion time, which the process spends above the threshold function.