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- W2192901128 abstract "Abstract . The numbers of level crossings and extremes for randomprocesses and fields play an important role in reliability theory and manyengineering applications. In many cases for Gaussian processes thePoisson approximation for their asymptotic distributions is used. Thispaper extends an approach proposed in Rusakov and Seleznjev (1988) forsmooth random processes on a finite interval. It turns out that a numberof functionals (including some integervalued ones) become continuous onthe space of smooth functions and weak convergence results for the se-quences of such continuous functionals are applicable. Examples of suchfunctionals for smooth random processes on infinite intervals and forrandom fields are studied. Key words: random process, extremes, level crossings, weak con-vergence, Gaussian processes AMS subject classifications: 60F17ReceivedAugust13,2001 AcceptedSeptember14,2001 1. Introduction The asymptotic behaviourofthe number oflevel crossingsor extremesfor a se-quenceofrandomfunctionshasbeenstudiedindetailmostlyfortheGaussiancase,andisnotverymuchknownfornon-Gaussianprocesses.ThemainlyusedapproachexploitedPoissonapproximationforthecorrespondingdistributions(seeforrefer-ences, e.g., Leadbetter, Lindgren, and Rootz´en(1983), Piterbarg(1996), H¨usler,Piterbarg,andSeleznjev(2000)).Forsuchproblems,RusakovandSeleznjev(1988)proposedanapproachbasedoncontinuitypropertiesofsomefunctionalsofinter-estforthesequenceofvectorrandomprocessesonaboundedinterval. Narrowingthe classof continuousfunctions" @default.
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- W2192901128 date "2001-12-20" @default.
- W2192901128 modified "2023-09-24" @default.
- W2192901128 title "On weak convergence of functionals on smooth random functions" @default.
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