Ramachandran (1969, Theorem 8) has shown that for any univariate infinitely divisible distribution and any positive real number $α$, an absolute moment of order $α$ relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding L$\acute{\rm e}$vy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, is given by Sato (1999, Theorem 25.3). We extend Ramachandran's theorem to the multivariate case, keeping in mind the immediate requirements under appropriate assumptions of cumulant studies of the distributions referred to; the format of Sato's theorem just referred to obviously varies from ours and seems to be having a different agenda. Also, appealing to a further criterion based on the L$\acute{\rm e}$vy measure, we identify in a certain class of multivariate infinitely divisible distributions the distributions that are self-decomposable; this throws new light on structural aspects of certain multivariate distributions such as the multivariate generalized hyperbolic distributions studied by Barndorff-Nielsen (1977) and others. Various points of relevance to the study are also addressed through specific examples.
A characterization of signed discrete infinitely divisible distributions
本文首先对负概率模型和拟无限可分性进行了综述。接下来，我们将Feller对离散无穷可分分布的刻画扩展到有符号离散无穷可分分布，这是一种与Lévy-Wiener定理有关的离散伪复合泊松分布。这是Sato（2014），Chaumont and Yor（2012）提出的一个开放问题的特例。对符号整数值无穷可分分布给出了一个涉及特征函数的类似结果。我们发现，许多分布是DPCP的非零p.g.f.性质，如混合泊松分布和分数泊松过程。DPCP有一些奇怪的属性，其中之一是DPCP类中的参数$λ$不能任意小。
