Characterization of the inverse stable subordinator

In this paper, we characterize the class of the inverse stable subordinator (E(t))t>0 by an independence property with a positive random variable T. Moreover, we extend this subordinator to a bivariate stochastic process ((E1(t),E2(t)))t>0 and we establish a characterization of this process using the notion of cut in natural exponential family and some independence conditions. This allows us to show that this extended process comes from a mixture between a β-stable process, with β∈(0,2] and an inverse α-stable subordinator, with α∈(0,1). We consider separately the case β=1 and the case β≠1.