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.
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This research was supported by a research scholarship from the Deutsche Forschungsgemeinschaft (DFG).
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Meerschaert, M.M., Scheffler, HP. Series representation for semistable laws and their domains of semistable attraction. J Theor Probab 9, 931–959 (1996). https://doi.org/10.1007/BF02214258
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DOI: https://doi.org/10.1007/BF02214258