Characterizations of multiparameter Cox and Poisson processes by the renewal property

In the first part of this paper, we give two characterizations of the multiparameter Poisson process: one characterization of the Poisson process uses the renewal property in the space R+d, and the other characterization uses the exponential distributions of random areas of rectangles. Finally, we obtain a partial generalization of a characterization of the random measure associated with a renewal Cox process from R+R+ to R+d.