A supplement to a theorem of Gut and Spa˘taru

Gut and Spa˘taru (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 <f>α</f>, <f>1<α&les;2</f>. 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. [Copyright &y& Elsevier]