We introduce a new class of distributions by generalizing the subexponential class to investigate the asymptotic relation between the tails of an infinitely divisible distribution and its Le ´vy measure. We call a one-sided distribution ì O-subexponential if it has positive tail satisfying lim sup x!1 ì Ã ì(x, 1)=ì(x, 1) , 1. Necessary and sufficient conditions for an infinitely divisible distribution to be O-subexponential are given in a similar way to the subexponential case in work by Embrechts et al. It is of critical importance that the O-subexponential is not closed under convolution roots. This property leads to the difference between our result and that corresponding to the subexponential class. Moreover, under the assumption that an infinitely divisible distribution has exponential tail, it is shown that an infinitely divisible distribution is convolution equivalent if and only if the ratio of its tail and its Le ´vy measure goes to a positive constant as x goes to infinity. Additionally, the upper and lower limits of the ratio of the tails of a semi-stable distribution and its Le ´vy measure are given.

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