Geodesic
On Asymptotic Properties of Some Statistics Similar to $\chi^2$
Teoriâ veroâtnostej i ee primeneniâ, Tome 1 (1956) no. 3, pp. 344-348

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A sequence of sequences of tests is considered (independent in each sequences) where possibleoutcomes $E_1,E_2,\dots,E_n$ have probabilities of $p_1,p_2,\dots,p_n$ respectively, where $p_i>0$ and $\sum_i p_i=1$. A group of possible outcomes $(E_1,E_2,\dots,E_n)$ is distinguished for which $$\lim_{N\to\infty}\max_{1\leq k\leq m}p_{i_k}=0,\text{ и }\sum_{k=1}^m p_{i_k}=\alpha_0,$$ where $m$ and $\alpha_0$ are independent of the number of sequences $N$. Theorems are given for sequences of sequences of certain statistics similar in structure to $\chi^2$, which show that these sequences converge to appropriate continuous Markov processes. 
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     author = {I. I. Gikhman},
     title = {On {Asymptotic} {Properties} of {Some} {Statistics} {Similar} to $\chi^2$},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {344--348},
     publisher = {mathdoc},
     volume = {1},
     number = {3},
     year = {1956},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1956_1_3_a3/}
}
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I. I. Gikhman. On Asymptotic Properties of Some Statistics Similar to $\chi^2$. Teoriâ veroâtnostej i ee primeneniâ, Tome 1 (1956) no. 3, pp. 344-348. http://geodesic.mathdoc.fr/item/TVP_1956_1_3_a3/