Limit theorems arising from sequences of multinomial random vectors. (English) Zbl 0669.60031
Set for independent and identically distributed random vectors \(X_ 1,...,X_ n\), \(X_ i\sim Mn(1,p_ 1,...,p_ k)\), \(K_ n=\sum^{n}_{i=1}X_ i\), \(S_ n=\sum^{n}_{i=1}iX_ i\). Then after normalizing \(S_ n-((n+1)/2)K_ n\) by suitable matrices the distribution converges to a multivariate normal distribution (Theorem 1).
If \(p_{\ell}=p_{\ell}^{(n)}\) and \(p_{\ell}^{(n)}\) is away from 0 and 1 for \(1\leq \ell \leq u\) whereas \(np_ j^{(n)}\to \lambda_ j>0\), \(j>u\), then after a suitable linear transformation of \(S_ n\) the distribution converges to a product distribution where the first distribution is a normal one and the second is a product of Poisson distributions.
If \(p_{\ell}=p_{\ell}^{(n)}\) and \(p_{\ell}^{(n)}\) is away from 0 and 1 for \(1\leq \ell \leq u\) whereas \(np_ j^{(n)}\to \lambda_ j>0\), \(j>u\), then after a suitable linear transformation of \(S_ n\) the distribution converges to a product distribution where the first distribution is a normal one and the second is a product of Poisson distributions.
Reviewer: L.Liese
Keywords:
multivariate Wilcoxon distribution; compound Poisson distribution; multivariate normal distributionReferences:
| [1] | Billingsley, P., (Convergence of Probability Measures (1968), Wiley: Wiley New York) · Zbl 0172.21201 |
| [2] | Chung, K. L., (A Course in Probability Theory (1974), Academic Press: Academic Press New York) · Zbl 0345.60003 |
| [3] | Gnedenko, B. V.; Kolmogorov, A. N., (Limit Distributions for Sums of Independent Random Variables (1954), Addison-Wesley: Addison-Wesley Cambridge, MA), (Translated by K. L. Chung) · Zbl 0056.36001 |
| [4] | Lehmann, E., (Nonparametrics: Statistical Methods Based on Ranks (1975), Holden-Day: Holden-Day San Francisco) · Zbl 0354.62038 |
| [5] | Panganiban, R., Limit Distributions of Test Statistics in the Analysis of Categorical Data, (Ph.D. Dissertation (1986), University of California: University of California Irvine) |
| [6] | Rvaceva, E. L., On domains of attractions of multi-dimensional distributions, Select. Transl. Math. Statist. Probab. IMS AMS, 2, 183-204 (1962) |
| [7] | Tucker, H. G., (A Graduate Course in Probability (1967), Academic Press: Academic Press New York) · Zbl 0159.45702 |
| [8] | Tucker, H. G., Joint limit laws of sample moments of a symmetric distribution, Ann. Probab., 8, 991-998 (1980) · Zbl 0441.60017 |
| [9] | Tucker, Hudson, and Veeh\(m\)Probab. Theory Relat. Fields; Tucker, Hudson, and Veeh\(m\)Probab. Theory Relat. Fields · Zbl 0659.60036 |
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