A supplement to a theorem of Gut and Sp˘ataru

Gut and Sp˘ataru (J. Math. Anal. Appl. 248 (2000) 233–246) proved a precise asymptotic theorem for random variables in the normal domain of attraction of some stable distribution with exponent α, 1<α 2. They also conjectured that the theorem should hold for the general case of the domain of attraction of a stable distribution. In this paper, we shall provide an example to deny their conjecture and study some further properties.  2003 Elsevier Inc. All rights reserved