Geodesic
The sequential chi-square test based on $s$-tuples of states of a Markov chain
Diskretnaya Matematika, Tome 9 (1997) no. 4, pp. 127-136.

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A multidimensional statistic whose components are one-dimensional chi-square statistics based on frequencies of outcomes of a sequence of multinomial trials was considered previously and the density of the limit distribution of this statistic was found. We extend this result to statistics based on $s$-tuples of states of a finite homogeneous Markov chain. The research was supported by the Russian Foundation for Basic Research, grant 96-01-00531. 
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     author = {B. I. Selivanov and V. P. Chistyakov},
     title = {The sequential chi-square test based on $s$-tuples of states of a {Markov} chain},
     journal = {Diskretnaya Matematika},
     pages = {127--136},
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     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1997_9_4_a11/}
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B. I. Selivanov; V. P. Chistyakov. The sequential chi-square test based on $s$-tuples of states of a Markov chain. Diskretnaya Matematika, Tome 9 (1997) no. 4, pp. 127-136. http://geodesic.mathdoc.fr/item/DM_1997_9_4_a11/