Multivariate stable exponential families and Tweedie scale

In this paper, we characterize the multivariate stable natural exponential families by a property of homogeneity of the cumulant function of some basis, and by a property of homogeneity of the variance function. We also extend the definition of a Tweedie scale to a finite dimensional space and we give a class of natural exponential families belonging to this scale on the space of symmetric matrices.