Sojourn measures of Student and Fisher-Snedecor random fields

Geometric characteristics of random surfaces play a crucial role in areas such as geoscience, environmetrics, astrophysics, and medical imaging, just to mention a few examples. Numerous real data have been modelled as Gaussian random processes or fields and studying of their excursion sets is now a well developed subject. Sojourn measures provide a classical approach to addressing various applied problems within this framework. There is a very rich literature on the topic, therefore below we cite only some key publications related to our approach. Good introductory references to some applications can be found in [2,6,14,36,38]. Sojourn measures of stochastic processes were studied extensively in a number of contexts and explicit formulae for their statistical characteristics were obtained for various scenarios, see, for example, [12,25,26], results for Gaussian stochastic processes with long range dependence in [8, 9], and also numerous references therein. Unfortunately, one cannot expect that the same will occur for the multidimensional situation. For random fields explicit formulae for the excursion distributions are rarely known, see [2,11]. Most published papers concern only first two moments of sojourn measures. However, it turned out that there are some interesting asymptotic results in this area. Such results are usually the main tools for statistical applications. It is natural to consider the volume of excursion sets in a bounded observation window and to study its limit behaviour as the window size grows. Some progress in this direction has been made in [1,14,29, 30,32,33,37]. The approach taken in the paper continues this line of investigations. The paper [14] studied central limit theorems for the volumes of excursion sets of stationary quasi-associated random fields and suggested two open problems: the extension of the results to different classes of random fields and the investigation of asymptotics for strongly dependent structures.