Extreme Value Theory for Continuous Parameter Stationary Processes.
NORTH CAROLINA UNIV AT CHAPEL HILL INST OF STATISTICS
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In this paper the central distributional results of classical extreme value theory are obtained, under appropriate dependence restrictions, for maxima of continuous parameter stochastic processes. In particular we prove the basic result here called Gnedenkos Theorem concerning the existence of just three types of non-degenerate limiting distributions in such cases, and give necessary and sufficient conditions for each to apply. The development relies, in part, on the corresponding known theory for stationary sequences. The general theory given does not finiteness of the number of upcrossings of any level x. However when the number per unit time is a.s. finite and has a finite mean mux, it is found that the classical criteria for domains of attraction apply when mux is used in lieu of the tail of the marginal distribution function. The theory is specialized to this case and applied to give the general known results for stationary normal processes for which mux may or may not be finite. A general Poisson convergence theorem is given for high level upcrossings, together with its implications for the asymptotic distributions of r-th local maxima. Author
- Statistics and Probability