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What portion of the sample makes a partial sum asymptotically stable or normal? (English) Zbl 0572.60028
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 $$0<\alpha \leq 2$$ 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.

##### MSC:
 60F05 Central limit and other weak theorems 60F15 Strong limit theorems 60J65 Brownian motion
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##### References:
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