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GENERALIZED MULTIVARIATE HERMITE DISTRIBUTIONS AND

RELATED POINT PROCESSES

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

^{1} *Department of Mathematics, University of Western Australia, Nedlands 6009, Australia*

^{2} *Division 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 { \sum_{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.

**Source**
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