Summary
LetX 1,X 2, ..., be a sequence of independent and identically distributed random variables in the domain of normal attraction of a nonnormal stabler law. It is known that only the sum of thek n largest andk n smallest extreme values in thenth partial sum withk n →∞ andk n /n→0 are responsible for the asymptotic stable distribution of the whole sum. We investigate the rate at which such sums of extreme values converge to a stable law in conjunction with the rate at which the sums of the middle terms become asymptotically negligible. In terms of rates of convergence our results provide in many cases a quantitative measure of exactly what portion of the sample is asymptotically stable.
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Research partially supported by the Deutsche Forschungsgemeinschaft while visiting the University of Delaware
Research partially supported by NSF Grant no. DMS-8803209
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Janssen, A., Mason, D.M. On the rate of convergence of sums of extremes to a stable law. Probab. Th. Rel. Fields 86, 253–264 (1990). https://doi.org/10.1007/BF01474645
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DOI: https://doi.org/10.1007/BF01474645
Keywords
- Stochastic Process
- Probability Theory
- Quantitative Measure
- Mathematical Biology
- Stable Distribution