In this paper we obtain an approximation for the duration distribution of the excursion set generated by the minimum of two independent F random processes above a high threshold u. Moreover, we obtain a closed-form approximation for the mean duration of the conjunction of these two F processes. As an illustration, we conduct a simulation study to compare the performances of the approximated distribution and the exact distribution.

We give explicit expressions for the Slepian model process of non-stationary Gaussian processes following level crossings and local maxima. We also include a detailed analysis of the high-level case.

In this paper, the behavior of a Gaussian random field near an ‘upcrossing' of a fixed level is investigated by strengthening the results of Wilson and Adler (1982) to full weak convergence in the space of functions which have continuous derivatives up to order 2. In Section 1, weak convergence and model processes are briefly discussed. The model field of Wilson and Adler (1982) is constructed in Section 2 using full weak convergence. Some of its properties are also investigated. Section 3 contains asymptotic results for the model field, including the asymptotic distributions of the Lebesgue measure of a particular excursion set and the maximum of the model field as the level becomes arbitrarily high.

In this paper, we investigate the behaviour of a Gaussian random field after an ‘upcrossing' of a particular level. In Section 1, we briefly discuss model processes and their background, and give a definition of an upcrossing of a level for random fields. A model field is constructed for the random field after an upcrossing of a level by using horizontal window conditioning in Section 2. The final section contains asymptotic distributions for the model field and for the location and height of the ‘closest' maximum to the upcrossing as the level becomes arbitrarily high.