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Limit theorems for multivariate discrete distributions. (English) Zbl 1092.60504
Summary: According to the usual law of small numbers a multivariate Poisson distribution is derived by defining an appropriate model for multivariate Binomial distributions and examining their behaviour for large numbers of trials and small probabilities of marginal and simultaneous successes. The weak limit law is a generalization of Poisson’s distribution to larger finite dimensions with arbitrary dependence structure. Compounding this multivariate Poisson distribution by a Gamma distribution results in a multivariate Pascal distribution which is again asymptotically multivariate Poisson. These Pascal distributions contain a class of multivariate geometric distributions. Finally the bivariate Binomial distribution is shown to be the limit law of appropriate bivariate hypergeometric distributions. Proving the limit theorems mentioned here as well as understanding the corresponding limit distributions becomes feasible by using probability generating functions.

##### MSC:
 60F05 Central limit and other weak theorems
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##### References:
 [1] Aitken AC (1936) A further note on multivariate selection. Proceedings of the Edinburgh Mathematical Society (Series 2) 5: 37–40 · Zbl 0015.22004 · doi:10.1017/S0013091500008245 [2] Aki S (1985) Discrete distributions of order k on a binary sequence. Annals of the Institute of Statistical Mathematics 37: 205–224 · Zbl 0577.62013 · doi:10.1007/BF02481092 [3] Beutler FJ (1983) A note on multivariate Poisson flows on stochastic processes. Advances in Applied Probability 15: 219–220. · Zbl 0504.60093 · doi:10.2307/1426994 [4] Bickel RJ, Freedman DJ (1981) Some asymptotic theory for the bootstrap. Annals of Statistics 9: 1196–1217. · Zbl 0472.62054 · doi:10.1214/aos/1176345637 [5] Block HW Savitz TH, Shaked M (1982) Some concepts of negative dependence. Annals of Probability 10: 765–772 · Zbl 0501.62037 · doi:10.1214/aop/1176993784 [6] Cacoullos T, Papageorgiou H (1983) Characterizations of discrete distributions by a conditional distribution and a regression function. Annals of the Institute of Statistical Mathematics 35: 95–104. · Zbl 0514.62018 · doi:10.1007/BF02480967 [7] Campbell JT (1934) The Poisson correlation function. Proceedings of the Edinburgh Mathematical Society (Series 2) 4: 18–26 · Zbl 0009.02801 · doi:10.1017/S0013091500024135 [8] Dawass M, Teicher H (1956) On infinitely divisible random vectors. Annals of Mathematical Statistics 27: 461–470 · Zbl 0078.31303 [9] Ehm W (1991) Binomial approximation to the Poisson binomial distribution. Stat. Probability Letters 11: 7–16 · Zbl 0724.60021 · doi:10.1016/0167-7152(91)90170-V [10] Famoye F, Consul PC (1995) Bivariate generalized Poisson distribution with some applications. Metrika 42: 127–139 · Zbl 0820.62015 · doi:10.1007/BF01894293 [11] Gordon FS, Gordon SP (1975) Transcendental functions of a vector variable and a characterization of a multivariate Poisson distribution. Statistical Distributions in Scientific Work 3: 163–172. [12] Gourieroux C, Monfort A (1979) On the characterization of a joint probability distribution by conditional distributions. Journal of Econometrics 10: 115–118 · doi:10.1016/0304-4076(79)90070-8 [13] Griffiths RC, Milne RK (1978) A class of bivariate Poisson processes. Journal of Multivariate Analysis 8: 380–395 · Zbl 0396.60050 · doi:10.1016/0047-259X(78)90061-1 [14] Griffiths RC, Milne RK, Wood R (1979) Aspects of correlation in bivariate Poisson distributions and processes. Australian Journal of Statistics 21: 238–255 · Zbl 0425.62038 · doi:10.1111/j.1467-842X.1979.tb01142.x [15] Hamadan MA (1973) A stochastic derivation of the bivariate Poisson distribution. South African Statistical Journal 7: 69–71 [16] Hamadan MA, Al-Bayyati HA (1971) Canonical expansion of the compound correlated bivariate Poisson distribution. Journal of the American Statistical Association 66: 390–393 · Zbl 0216.47904 · doi:10.2307/2283942 [17] Hawkes AG (1972) A bivariate exponential distribution with applications to reliability. Journal of the Royal Statistical Society, SeriesB, 34: 129–121 · Zbl 0255.60064 [18] Holgate P (1964) Estimation for the bivariate Poisson distribution. Biometrika 51: 241–245 · Zbl 0133.11802 [19] Jamardan KG (1973) Chance mechanisms for multivariate hypergeometric models. Sankhya A 35: 465–478 · Zbl 0327.62032 [20] Jamardan KG (1975) Certain inference problems for multivariate hypergeometric distributions. Communications in Statistics 4: 375–388 · Zbl 0301.62039 · doi:10.1080/03610927508827254 [21] Jamardan KG (1976) Certain estimation problems for multivariate hypergeometric models. Annals of the Institute of Statistical Mathematics 28: 429–444 · Zbl 0362.62069 · doi:10.1007/BF02504760 [22] Jensen DR (1971) A note on Positive dependence and the structure of bivariate distributions. SIAM Journal of Applied Mathematics 20: 749–752 · Zbl 0226.60024 · doi:10.1137/0120073 [23] Jogedo K (1968) Characterizations of independence in certain families of bivariate and multivariate distributions. Annals of Mathematical Statistics 39: 433–441 · Zbl 0164.49106 · doi:10.1214/aoms/1177698407 [24] Jogedo K (1975) Dependence concepts and probability inequalities. Statistical Distributions in Scientific Work 1: 271–279 [25] Jogedo K, Patil GP (1975) Probability inequalities for certain multivariate discrete distributions. Sankhya B, 37: 158–164. [26] Johnson N, Kotz S (1969) Distributions in statistics I: Discrete distributions. Houghton Mifflin Boston · Zbl 0292.62009 [27] Johnson N, Kotz S (1972) Distributions in statistics IV: Continuous multivariate distributions. John Wiley, New York. · Zbl 0248.62021 [28] Khatri CG (1971) On multivariate contagious distributions. Sankhya B, 33: 197–216 [29] Kocherlakota S, Kocherlakota K (1993) Bivariate discrete distributions. Marcel Dekker, New York · Zbl 0794.62002 [30] Krishnamoorthy AS (1951) Multivariate binomial and Poisson distributions. Sankhya 11: 117–124 · Zbl 0044.13502 [31] Krummenauer F (1996) Grenzwertsätze für multivariate diskrete Verteilungen. Doctoral Thesis in Statistics, University of Dortmund, Germany · Zbl 0874.62011 [32] Lukacs E (1979) Some multivariate statistical characterization theorems. Journal of Multivariate Analysis 9: 278–287 · Zbl 0409.62036 · doi:10.1016/0047-259X(79)90085-X [33] Lukacs E, Beer S (1977) Characterization of the multivariate Poisson distribution. Journal of Multivariate Analysis 7: 1–12 · Zbl 0358.62012 · doi:10.1016/0047-259X(77)90027-6 [34] Marshall AW, Olkin I (1985) A family of bivariate distributions generated by the bivariate Bernoulli distribution. Journal of the American Statistical Association 80: 332–338. · Zbl 0575.60023 · doi:10.2307/2287890 [35] Mitchell CR, Paulson AS (1981) A new bivariate negative binomial distribution. Naval Research Logistics Quaterly 28: 359–374 · Zbl 0463.60024 · doi:10.1002/nav.3800280302 [36] Nelson JF (1985) Multivariate Gamma-Poisson models. Journal of the American Statistical Association 8: 828–834 · doi:10.2307/2288540 [37] Nevill AM, Kemp CD (1975) On characterizing the hypergeometric and multivariate hypergeometric distributions. Statistical Distributions in Scientific Work 3: 353–358 [38] Panaretos J (1980) A characterization of a general class of multivariate discrete distributions. In: Gyeres (ed.) Analytic function methods in probability theory Colloquia Mathematica Societatis Janos Bolyai, 21, North Holland, Amsterdam, pp. 243–252 [39] Panaretos J (1983) An elementary characterization of the multinomial and the multivariate hypergeometric distribution. In: Kalashnikov VV, Zolotarev, VM (eds) Statistical problems for stochastic models. Springer-Verlag, New York, pp. 156–164. [40] Patil GP (1964) On certain compound Poisson and compound binomial distributions. Sankhya A 26: 293–294 · Zbl 0137.36601 [41] Sibuya M (1983) Generalized hypergeometric distributions. Encyclopedia of Statistical Sciences 3: 330–334 [42] Steyn HS (1951) On discrete multivariate probability functions of hypergeometric type. Koninklijke Nederlandse Akademie Wetenschappen Proceedings (Series A) 54: 23–30. · Zbl 0043.33901 [43] Steyn HS (1955) On discrete multivariate probability functions of hypergeometric type. Koninklijke Nederlandse Akademie Wetenschappen Proceedings (Series A) 58: 588–595 · Zbl 0068.12701 [44] Takeuchi K, Takemura A (1987) On sums of 0–1 random variables II: Multivariate case. Annals of the Institute of Statistical Mathematics 39, Part A: 307–324 · Zbl 0658.60031 · doi:10.1007/BF02491470 [45] Talwalker S (1970) A characterization of the double Poisson distribution. Sankhya A, 32: 265–270 · Zbl 0234.60011 [46] Teicher H (1954) On the multivariate Poisson distribution. Scandinavisk Aktuarietidskrift 37: 1–9 · Zbl 0056.35801 [47] Wang YH (1989) A multivariate extension of Poisson’s theorem. Canadian Journal of Statistics 17: 241–245 · Zbl 0691.60019 · doi:10.2307/3314853 [48] Wicksell SD (1923) Contributions to the analytical theory of sampling. Arkiv for Matematik, Astronomi Och Fysik 17: 1–46 · JFM 49.0727.02 [49] Zolotarev VM (1966) A multidimensional analogue of the Berry/Esséen inequality for sets with bounded diameter. SIAM Theory of Probability and Its Applications 11: 447–454. · doi:10.1137/1111045
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