Geodesic
Multivariate chi-square distribution for a nonhomogeneous multinomial scheme
Diskretnaya Matematika, Tome 10 (1998) no. 2, pp. 52-61

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We consider a statistic whose components are the $\chi^2$ statistics, constructed on the base of the frequencies of outcomes of non-homogeneous polynomial samples of growing volumes. Conditions are formulated which guarantee the existence of a limiting distribution of this statistics, and its Laplace transform is presented. In the homogeneous case, the Laplace transform obtained is reduced to the form which has been known before.This work was supported by the Russian Foundation for Basic Research, grants 96–01–00531, 96–15–96092. 
@article{DM_1998_10_2_a3,
     author = {B. I. Selivanov and V. P. Chistyakov},
     title = {Multivariate chi-square distribution for a nonhomogeneous multinomial scheme},
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     pages = {52--61},
     publisher = {mathdoc},
     volume = {10},
     number = {2},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1998_10_2_a3/}
}
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B. I. Selivanov; V. P. Chistyakov. Multivariate chi-square distribution for a nonhomogeneous multinomial scheme. Diskretnaya Matematika, Tome 10 (1998) no. 2, pp. 52-61. http://geodesic.mathdoc.fr/item/DM_1998_10_2_a3/