Generalized multivariate Hermite distributions and related processes

This paper is primarily concerned with the problem of characterizing those functions of the form[GRAPHICS]where z = [z1,..., z(n)]', which are probability generating functions. The corresponding distributions are called generalized multivariate Hermite distributions. Use is made of results of Cuppens (1975), with particular interest attaching to the possibility of some of the coefficients ak being negative.The paper goes on to discuss related results for point processes. The point process analogue of the above characterization problem was raised by Milne and Westcott (1972). This problem is not solved but relevant examples are presented. Ammann and Thall (1977) and Waymire and Gupta (1983) have established a related characterization result for certain infinitely divisible point processes. Their results are considered from a probabilistic viewpoint.