Series representation for semistable laws and their domains of semistable attraction |
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Authors: | Mark M Meerschaert Hans-Peter Scheffler |
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Institution: | (1) Department of Mathematics, University of Nevada, 89557 Reno, Nevada;(2) Present address: Fachbereich Mathematik, Universität Dortmund, Dortmund, Germany |
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Abstract: | If the centered and normalized partial sums of an i.i.d. sequence of random variables converge in distribution to a nondegenerate limit then we say that this sequence belongs to the domain of attraction of the necessarily stable limit. If we consider only the partial sums which terminate atk
n
wherek
n+1
ck
n
then the sequence belongs to the domain of semistable attraction of the necessarily semistable limit. In this paper, we consider the case where the limiting distribution is nonnormal. We obtain a series representation for the partial sums which converges almost surely. This representation is based on the order statistics, and utilizes the Poisson process. Almost sure convergence is a useful technical device, as we illustrate with a number of applications.This research was supported by a research scholarship from the Deutsche Forschungsgemeinschaft (DFG). |
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Keywords: | LePage series representation semistable laws domains of semistable attraction regular variation order statistics Poisson process trimmed sums self-normalized sums |
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