Local Subexponentiality and Self-decomposability

The class of exponential tilts of convolution equivalent distributions is determined. As a corollary, the local subexponentiality of one-sided infinitely divisible distributions is characterized. It is applied to the subexponentiality of the densities of a self-decomposable distribution and its Lévy measure. Bondesson’s conjecture on the density of the Lévy measure of a lognormal distribution is solved as an example. Results of Denisov et al. on the distributions of random sums are extended to the two-sided case. Finally, the local subexponentiality of the distribution of the supremum of a random walk is characterized.