### Infinite divisibility and generalized subexponentiality

Takaaki Shimura and Toshiro Watanabe
Source: Bernoulli Volume 11, Number 3 (2005), 445-469.

#### Abstract

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 Lévy measure. We call a one-sided distribution μ O-subexponential if it has positive tail satisfying $\limsup_{x \to \infty} \mu * \mu(x, \infty)/ \mu(x, \infty) < \infty$ . 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 Lévy 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 Lévy measure are given.

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Permanent link to this document: http://projecteuclid.org/euclid.bj/1120591184
Mathematical Reviews number (MathSciNet): MR2146890
Digital Object Identifier: doi:10.3150/bj/1120591184
Zentralblatt MATH identifier: 1081.60016

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