Geodesic
On repetitions of long tuples in a Markov chain
Diskretnaya Matematika, Tome 26 (2014) no. 3, pp. 79-89

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Let $X_0,X_1,\ldots$ be a simple ergodic finite Markov chain. We prove limit theorems for the distribution of the number $\tilde\xi(s,n)$ of events $$\{X_{i-1}\ne X_{j-1},\ X_{i+k}= X_{j+k},\ k=0,\ldots,s-1\},\quad 1\le i\le n,$$ when $s,n\to\infty$. Limit theorems for distributions of some random variables connected with $\tilde\xi(s,n)$ are derived as corollaries.
Keywords: Markov chain, repetitions of tuples, Poisson limit theorem. 
@article{DM_2014_26_3_a6,
     author = {V. G. Mikhailov and A. M. Shoitov},
     title = {On repetitions of long tuples in a {Markov} chain},
     journal = {Diskretnaya Matematika},
     pages = {79--89},
     publisher = {mathdoc},
     volume = {26},
     number = {3},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2014_26_3_a6/}
}
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V. G. Mikhailov; A. M. Shoitov. On repetitions of long tuples in a Markov chain. Diskretnaya Matematika, Tome 26 (2014) no. 3, pp. 79-89. http://geodesic.mathdoc.fr/item/DM_2014_26_3_a6/