Characterization of multivariate stable processes

This paper deals with a characterization of a multivariate stable process using an independence property with a positive random variable. Moreover, we establish a characterization of a multivariate Lévy process based on the notion of cut in a natural exponential family. This allows us to draw some related properties. More precisely, we give the probability density function of this process and the law of the mixture of the Lévy process governed by the convolution semigroup with respect to an exponential random variable. These results are confidentially connected with the univariate case given by [G. Letac and V. Seshadri, Exponential stopping and drifted stable processes, Stat. Probab. Lett ., 72:137–143, 2005].