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Point Processes Associated with Extreme Value Theory.
NORTH CAROLINA UNIV AT CHAPEL HILL CENTER FOR STOCHASTIC PROCESSES
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This work demonstrates the application of point process theory in the context of statistical extremes. Consider a stationary random sequence which satisfies certain dependence restrictions. We study the asymptotic behavior of a sequence of point processes that record the positions at which extreme values occur. Necessary and sufficient conditions are given for the weak convergence of the sequence. It is found that the usual Poisson limit when the random sequence is i.i.d. is replaced by a Compound Poisson limit. The asymptotic distributions of extreme order statistics are derived from the weak convergence result using simple combinatorial arguments. A class of point processes in two dimensions is also considered. The weak limit is characterized to be a cluster process which is determined by a homogeneous Poisson Process and the local dependence structure of the random sequence. A random sequence whose members are the weighted maxima of i.i.d. random variables is studied. It is shown that the sequence satisfies our dependence restrictions, and the point process results developed can be applied. Specific limit forms of the various point processes of interest are derived. Author
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