Geodesic
Limit Distributions of the $\chi^2$ Statistic of K.~Pearson in a Sequence of Independent Trials
Matematičeskie zametki, Tome 83 (2008) no. 6, pp. 899-911.

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We study the $\chi^2$ statistic of K. Pearson in a sequence of independent and, generally, inhomogeneous trials with a fixed number of outcomes. It is assumed that the probabilities of occurrence of outcomes of the trials satisfy certain conditions. This problem statement embraces familiar results for the $\chi^2$ statistic in the case of multinomial trials. We obtain explicit expressions and estimates for the expectation and the variance of the $\chi^2$ statistic. For the $\chi^2$ statistic centered and normalized in a suitable way, we find limit distributions (the normal one, the distribution of the sum of the squares of normal random variables and, in particular, the $\chi^2$ distribution). Conditions for the convergence to the corresponding limit distributions are given.
Keywords: $\chi^2$ statistic of K. Pearson, $\chi^2$ distribution, goodness-of-fit test, multinomial trials, (in)homogenous trials, asymptotically normal random variable.
Mots-clés : normal distribution 
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     author = {B. I. Selivanov},
     title = {Limit {Distributions} of the $\chi^2$ {Statistic} of {K.~Pearson} in a {Sequence} of {Independent} {Trials}},
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B. I. Selivanov. Limit Distributions of the $\chi^2$ Statistic of K.~Pearson in a Sequence of Independent Trials. Matematičeskie zametki, Tome 83 (2008) no. 6, pp. 899-911. http://geodesic.mathdoc.fr/item/MZM_2008_83_6_a8/

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