We show the equivalence of three properties for an infinitely divisible distribution: the subexponentiality of the density, the subexponentiality of the density of its L\'evy measure and the tail equivalence between the density and its L\'evy measure density, under monotonic-type assumptions on the L\'evy measure density. The key assumption is that tail of the L\'evy measure density is asymptotic to a non-increasing function or is eventually non-increasing. Our conditions are novel and cover a rather wide class of infinitely divisible distributions. Several significant properties for analyzing the subexponentiality of densities have been derived such as closure properties of [ convolution, convolution roots and asymptotic equivalence ] and the factorization property. Moreover, we illustrate that the results are applicable for developing the statistical inference of subexponential infinitely divisible distributions which are absolutely continuous.

翻译：我们证明了一个无限可分分布的三个性质之间的等价性：密度的次指数性、其Lévy测度密度的次指数性，以及密度与其Lévy测度密度之间的尾部等价性，这些结论是在对Lévy测度密度施加单调型假设的条件下获得的。关键假设是Lévy测度密度的尾部渐近于一个非增函数，或者最终是非增的。我们的条件是新颖的，并涵盖了相当广泛的无限可分分布类。我们推导出若干用于分析密度次指数性的重要性质，例如[卷积、卷积根与渐近等价性]的封闭性质，以及分解性质。此外，我们阐明了这些结果可应用于开发绝对连续次指数无限可分分布的统计推断。