Sojourn Times and the Fragility Index

Details Sojourn Times and the Fragility Index Sojourn Times and the Fragility Index

2011-07-28

arxiv.org/abs/1107.5696v2

Mathematics

Probability

22 pages

Scientific paper

10.1016/j.spa.2011.11.009

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.

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