Summary
This paper applies the stochastic calculus of multiple Wiener-Itô integral expansions to express the number of crossings of the mean level by a stationary (discrete- or continuous-time) Gaussian process within a fixed time interval [0,T]. The resulting expansions involve a class of hypergeometric functions, for which recursion and differential relations and some asymptotic properties are derived. The representation obtained for level-crossing counts is applied to prove a central limit theorem of Cuzick (1976) for level crossings in continuous time, using a general central limit theorem of Chambers and Slud (1989a) for processes expressed via multiple Wiener-Itô integral expansions in terms of a stationary Gaussian process. Analogous results are given also for discrete-time processes. This approach proves that the limiting variance is strictly positive, without additional assumptions needed by Cuzick.
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Research supported by Office of Naval Research contracts N00014-86-K-0007 and N00014-89-J-1051
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Slud, E. Multiple Wiener-itô integral expansions for level-crossing-count functionals. Probab. Th. Rel. Fields 87, 349–364 (1991). https://doi.org/10.1007/BF01312215
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DOI: https://doi.org/10.1007/BF01312215