On excursion probabilities for a class of non-stationary random fields

Asymptotic approximations of threshold exceedance probabilities for random fields have been established, under suitable conditions, in terms of connectivity properties defined by the expectation of Euler characteristic. In this paper, some extensions and related results concerning the order of approximation are investigated for the class of harmonizable random fields. In particular, the study is focused on transformations of stationary random fields by spatial deformations subject to appropriate regularity and boundedness assumptions. Other well-known significant cases of practical interest within this class are also addressed. Finally, several aspects of continuing research in this context are discussed.