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A note on Poisson approximation in multivariate case. (English) Zbl 0633.60034
In recent years has been discussed the Poisson approximation for the sum of independent Bernoulli random vectors. We usually define multivariate Poisson distributions P(\(\lambda)\) by \(P(x=K)=\sum_{[C]}\prod_{i\in E}p(\alpha_ i;\lambda_ i)\) where \(p(\alpha_ i;\lambda_ i)\) is a univariate Poisson density.
K. Kawamura [ibid. 2, 337-345 (1979; Zbl 0434.60019)] has derived sufficient conditions of a Poisson approximation to the sum of independent identically multivariate Bernoulli random vectors. The author has discussed the multivariate Poisson distribution by the limiting value of the sum of Bernoulli random vectors and shown sufficient conditions for the Poisson approximation to the sum of independent Bernoulli random vectors which may or may not be identically distributed.
The converse assertion of this paper will also be true which needs very complicated proofs and will be published in the near future, that is, the conditions are also necessary for the Poisson approximation. Finally, this paper gives a trivial result if we suppose that the sequence of probability distributions has the property of “smallness”, that is, \(\lim_{k}\min_{j}P(X_{kj}=0)=1\).

MSC:
60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
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[1] M. POLAK, Poisson approximation for sums of independent bivariate Bernoulli vectors, Kodai Math. J., 5 (1982), 408-415. · Zbl 0522.60026 · doi:10.2996/kmj/1138036608
[2] C. Liu, Necessary and sufficient conditions for a Poisson approximation (trivariat case), kodai Math. J. 9(3) (1986), 368-384. · Zbl 0633.60033 · doi:10.2996/kmj/1138037265
[3] K. KAWAMURA, The structure of multivariate Poisson distribution, Kodai Mat J. 2(3) (1979), 337-345. · Zbl 0434.60019 · doi:10.2996/kmj/1138036064
[4] A. W. MARSHALL AND I. OLKIN, A family of bivariate distributions generated b the bivariate Bernoulli distribution, Journal of the A. S. A., (June 1985), 332-338. · Zbl 0575.60023 · doi:10.2307/2287890
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