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We show that all possible moderately trimmed sums from a domain of geometric partial attraction of a semistable law G are asymptotically normal, along the whole sequence of natural numbers, under the necessary condition that the Levy functions of G do not have flat stretches. We also show that asymptotic normality prevails still along the whole sequence at least for suitably chosen moderately trimmed sums from such a domain for every G. It then follows that after removing any moderately trimmed sum from the middle the remaining sums of extreme values still produce every semistable limiting distribution G that the original full sums have, along exactly the same geometrically growing subsequences of the natural numbers.

Trimmed sums from the domain of geometric partial attraction of semistable laws