Summary
Let a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index 0<α≦2 be given. We show that if at each stage n a number k n depending on n of the lower and upper order statistics are removed from the n-th partial sum of the given random variables then under appropriate conditions on k n the remaining sum can be normalized to converge in distribution to a standard normal random variable. A further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour. Our main tool is a new Brownian bridge approximation to the uniform empirical process in weighted supremum norms.
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Work done while visiting the Bolyai Institute, Szeged University, partially supported by a University of Delaware Research Foundation Grant
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Csörgő, S., Horváth, L. & Mason, D.M. What portion of the sample makes a partial sum asymptotically stable or normal?. Probab. Th. Rel. Fields 72, 1–16 (1986). https://doi.org/10.1007/BF00343893
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DOI: https://doi.org/10.1007/BF00343893