GENERALIZED MULTIVARIATE HERMITE DISTRIBUTIONS AND RELATED POINT PROCESSES

R. K. MILNE^{1} AND M. WESTCOTT^{2}

1Department of Mathematics, University of Western Australia, Nedlands 6009, Australia

2Division of Mathematics and Statistics, CSIRO, P.O. Box 1965, Canberra City 2601, Australia

(Received July 15, 1990; revised June 19, 1992)

Abstract. This paper is primarily concerned with the problem of characterizing those functions of the form

G(z ) = exp { ∑_{0≤k′1≤m} a_{k}(z^{k} − 1)} ,

where z = [z_{1}, …, 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 a_{k} 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.

Key words and phrases: Hermite distribution, generalized multivariate Hermite distribution, point process, probability generating function.