On multiple-level excursions by stationary processes with deterministic peaks

A well-known property of stationary Gaussian processes is that the excursions over high levels ("peaks") have a limiting parabolic shape, each determined by a single random parameter. This means, in particular, that (in the limit) the length of a single excursion above a high level determines the length of the (shorter) excursion above each higher level. In this paper we consider a general class of stationary processes with this property. Results of Leadbetter and Hsing (1990) for convergence of exceedance random measures are generalized to include multiple-level exceedances and developed further for the above class of processes. Specific application is made to stationary normal processes.