Extremes and upcrossing intensities for P-differentiable stationary processes

Given a stationary differentiable in probability process {ξ(t)}t∈R we express the asymptotic behaviour of the tail P{supt∈[0,1] ξ(t)>u} for large u through a certain functional of the conditional law (ξ′(1)|ξ(1)>u). Under technical conditions this functional becomes the upcrossing intensity μ(u) of the level u by ξ(t). However, by not making explicit use of μ(u) we avoid the often hard-to-verify technical conditions required in the calculus of crossings and to relate upcrossings to extremes. We provide a useful criterion for verifying a standard condition of tightness-type used in the literature on extremes. This criterion is of independent value. Although we do not use crossings theory, our approach has some impact on this subject. Thus we complement existing results due to, e.g. Leadbetter (Ann. Math. Statist. 37 (1983) 260–267) and Marcus (Ann. Probab. 5 (1977) 52–71) by providing a new and useful set of technical conditions which ensure the validity of Riceʹs formula μ(u)=∫0∞ zfξ(1),ξ′(1)(u,z) dz. As examples of application we study extremes of Rn-valued Gaussian processes with strongly dependent component processes, and of totally skewed moving averages of α-stable motions. Further we prove Belayevʹs multi-dimensional version of Riceʹs formula for outcrossings through smooth surfaces of Rn-valued α-stable processes.