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Merging of linear combinations to semistable laws. (English) Zbl 1173.60306
Summary: We prove merge theorems along the entire sequence of natural numbers for the distribution functions of suitably centered and normed linear combinations of independent and identically distributed random variables from the domain of geometric partial attraction of any non-normal semistable law. Surprisingly, for some sequences of linear combinations, not too far from those with equal weights, the merge theorems reduce to ordinary asymptotic distributions with semistable limits. The proofs require working out general conditions for merging in terms of characteristic functions.

60E07Infinitely divisible distributions; stable distributions
60E05General theory of probability distributions
60E10Transforms of probability distributions
60F05Central limit and other weak theorems
Full Text: DOI
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