Moment properties of multivariate infinitely divisible laws and criteria for self-decomposability

Ramachandran (1969, Theorem 8) has shown that for any univariate infinitely divisible distribution and any positive real number $\alpha$, an absolute moment of order $\alpha$ 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.