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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.

60F05 Central limit and other weak theorems
60F15 Strong limit theorems
60J65 Brownian motion
Full Text: DOI
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