The authors describe the scenarios for the convergence of the (N +1)-dimensional Pólya distribution in the domain of “rare events” area to the N-dimensional α-generalized Poisson distribution in a wide range of the parameters (n, \( \overline{p} \) = (p1, . . . , pN), α) and arguments (\( \overline{k} \) = (k1, . . . , kN)) of the Pólya distribution. Also, minimal constraints, natural for problems related with the limit behavior of the P´olya distribution (or Pólya random walks), are stipulated on parameter α (nα ∈\( \left(-\frac{1}{2},1\right) \)). The rate of approach of the multidimensional Pólya distribution to the product of one-dimensional Pólya distributions with parameters n, pm, and α (m = 1, . . . ,N) is estimated and the asymptotic estimates are presented for the mathematical expectations of functions of components of the random vector \( \overline{X} \) , which has the multidimensional Pólya distribution.
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References
K.G. Janardan and G.P. Patil, “A unified approach for a class of multivariate hypergeometric models,” Sankhya. Ser. A, 34, No. 4, 363–376 (1972).
Ya. P. Lumelskiy,“Random walks corresponding to generalized urn schemes,” Dokl. AS USSR, 209, No. 6, 1281–1284 (1973).
Ya. P. Lumelskiy and Ya.M. L’vovskiy, “Asymptotic formulas for multidimensional P´olya distributions,” Dokl. AS USSR, 246, No. 3, 536–540 (1979).
Ya.M. L’vovskiy and E.G. Tsylova, “Proof of limit theorems for the Pólya distribution with the help of generalized Appell polynomials,” Statist. Meth. Estim. Hypot. Test., 162–169 (1984).
E.G. Tsylova, “Generalized Poisson walks,” in: Materials of the V USSR Workshop on Nonparametric and Rrobust Statistics Methods in Cybernetics (September, 1985), Part II, Krasnoyarsk–Tomsk (1985), pp. 326–335.
E.G. Tsylova, The α-generalized Poisson distribution, Deposited in VINITI, No. 7794–85 (1985).
E.G. Tsylova, “Generalized Poisson distribution generated by the α-generalized Poisson walk,” Math. Statist. Appl., 204–216 (1987).
E.G. Tsylova, “Multidimensional α-generalized Poisson random walks,” Statist. Meth. Estim. Hypot. Test., No. 9, 177–186 (1993).
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Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, Vol. 22, pp. 129–136, 2010
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Tsylova, E., Ekgauz, E. Refinement of Asymptotic Formulas for the Multidimensional Pólya Distribution in the Domain of “Rare Events”. J Math Sci 267, 51–56 (2022). https://doi.org/10.1007/s10958-022-06106-1
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DOI: https://doi.org/10.1007/s10958-022-06106-1