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A note on the recurrent relations for the bivariate Poisson distribution. (English) Zbl 0574.60024
It is a well known fact that a univariate Poisson density \(p(k;\lambda) = (\lambda^ k/k!) \exp(-\lambda)\) \((k=0,1,2,...)\) satisfies a relation \(k\;p(k;\lambda) = \lambda p(k-1;\lambda)\) \((k\geq 1)\). And also, we know recurrence relations for a bivariate Poisson density. The relations have been expressed as (1) \(k\;p(k,l) = \lambda_{10}p(k-1,l) + \lambda_{11}p(k-1,l-1)\) and (2) \(l\;p(k,l) = \lambda_{01}p(k,l-1) + \lambda_{11}p(k-1,l-1)\) where \(p(k,l)\) \((k,1=0,1,2,...)\) is a density of a bivariate Poisson distribution and \(\lambda_{10}\), \(\lambda_{01}\) and \(\lambda_{11}\) are its nonnegative parameters.
In this paper, the author asserts that the bivariate Poisson density satisfies the relations (1) and (2) for \(k,l=0,1,2,..\). and in the case of at least one of k and l equals zero, (3) \(k\;p(k,0) = \lambda_{10}p(k-1,0)\) for \(k=0,1,2,...\), (4) \(l\;p(0,l) = \lambda_{01}p(0,l-1)\) for \(k=0,1,2,..\). and (5) \(p(0,0) = \exp(-\lambda_{10}-\lambda_{01}-\lambda_{11})\). He also asserts that, conversely, if we assume the relations (1)-(5) for a nonnegative function \(p(k,l)\), we can calculate all the values of \(p(k,l)\) for every nonnegative integers \(k\) and \(l\).
The author’s main assertion is that the recurrence relations (1)-(5) are self duplicated: more precisely, it should be expressed as; the main relations (1) and (2) are self duplicated to calculate the density. The recurrence relations should be rearranged as minimum relations as to calculate the density \(p(k,l)\), that is: [(1) or (2)] and (3), (4) and (5) are the sufficient relations to calculate the density, conversely.
The assertion of this paper has already been generalized to the multivariate case and will be published in the same journal. The title of this paper is somewhat incorrect in using the word ”recurrent” which should better be replaced by ”recurrence”.

MSC:
60E99 Distribution theory
62E10 Characterization and structure theory of statistical distributions
60G50 Sums of independent random variables; random walks
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[1] FELLER, W., An introduction to probability theory and its applications, Vol. 1, sec. ed. 6th print (1961).
[2] KAWAMURA, K., The structure of bivariate Poisson distribution, Kdai Math. Sem. Rep., 25(2) (1973), 246-256. · Zbl 0266.60004 · doi:10.2996/kmj/1138846776
[3] JOHNSON, Norman L. AND KOTZ Samuel, Discrete distributions, Houghton Mifflin Co.(1969). · Zbl 0292.62009
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