Abstract
This paper is primarily concerned with the problem of characterizing those functions of the form
wherez=[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 coefficientsa 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.
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References
Ammann, L. P. and Thall, P. F. (1977). On the structure of regular infinitely divisible point processes,Stochastic Process. Appl.,6, 87–94.
Ammann, L. P. and Thall, P. F. (1978). Count distributions, orderliness and invariance of Poisson cluster processes,J. Appl. Probab.,16, 261–273.
Cuppens, R. (1975).Decomposition of Multivariate Probabilities, Academic Press, New York.
Daley, D. J. and Vere-Jones, D. (1988).An Introduction to the Theory of Point Processes, Springer, Berlin.
Feller, W. (1968).An Introduction to Probability Theory and Its Applications, Volume 1, 3rd ed., Wiley, New York.
Gupta, R. P. and Jain, G. C. (1974). A generalized Hermite distribution and its properties,SIAM J. Appl. Math.,27, 359–363.
Kallenberg, O. (1975).Random Measures, Akademie-Verlag, Berlin (Also (1976) Academic Press, London).
Kemp, A. W. and Kemp, C. D. (1965). Some properties of the ‘Hermite’ distribution,Biometrika,52, 381–394.
Kemp, C. D. and Papageorgiou, H. (1982). Bivariate Hermite distributions,Sankhyā Ser. A,44, 269–280.
Kerstan, J. and Matthes, K. (1964). Stationäre zufällige Punktfolgen II,Jahresbericht der Deutschen Mathematiker-Vereinigung,66, 106–118.
Lévy, P. (1937). Sur les exponentielles des polynômes et sur l'arithmétique des produits des lois de Poisson,Ann. Sci. École Norm. Sup.,54, 231–292.
Lukacs, E. (1970).Characteristic Functions, 2nd ed. Griffin, London.
Matthes, K. (1963). Unbeschränkt teilbare Verteilungsgesetze stationärer Punktfolgen,Wissenschaftliche Zeitschrift der Hochschule für Elektrotechnik Ilmenau,9, 235–238.
Matthes, K., Kerstan, J. and Mecke, J. (1978).Infinitely Divisible Point Processes, Wiley, Chichester.
Milne, R. K. and Westcott, M. (1972). Further results for Gauss-Poisson processes,Adv. in Appl. Probab.,4, 151–176.
Moyal, J. E. (1958). Discussion on the paper of Neyman and Scott (1958), 36–37.
Moyal, J. E. (1962). The general theory of stochastic population processes,Acta Math.,108, 1–31.
Newman, D. S. (1970). A new family of point processes which are characterized by their second moment properties.J. Appl. Probab.,7, 338–358.
Neyman, J. and Scott, E. L. (1958). A statistical approach to problems of cosmology (with discussion),J. Roy. Statist. Soc. Ser. B,20, 1–43.
Shanbhag, D. N. and Westcott, M. (1977). A note on infinitely divisible point processes,J. Roy. Statist. Soc. Ser. B,39, 331–332.
Steyn, H. S. (1976). On the multivariate Poisson Normal distribution,J. Amer. Statist. Assoc.,71, 233–236.
Szegö, G. (1939).Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, New York.
Teicher, H. (1954). On the multivariate Poisson distribution,Skand. Aktuarietidskr.,37, 1–9.
Van Harn, K. (1978). Classifying infinitely divisible distributions by functional equations, Mathematical Centre Tract 103, Mathematisch Centrum, Amsterdam.
Waymire, E. and Gupta, V. K. (1983). An analysis of the Polya point process,Adv. Appl. Probab.,15, 39–53.
Westcott, M. (1971). On existence and mixing results for cluster point processes,J. Roy. Statist. Soc. Ser. B,33, 290–300.
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Milne, R.K., Westcott, M. Generalized multivariate Hermite distributions and related point processes. Ann Inst Stat Math 45, 367–381 (1993). https://doi.org/10.1007/BF00775822
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DOI: https://doi.org/10.1007/BF00775822