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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - V. G. Mikhailov
AU  - A. M. Shoitov
TI  - On repetitions of long tuples in a Markov chain
JO  - Diskretnaya Matematika
PY  - 2014
SP  - 79
EP  - 89
VL  - 26
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DM_2014_26_3_a6/
LA  - ru
ID  - DM_2014_26_3_a6
ER  - 
%0 Journal Article
%A V. G. Mikhailov
%A A. M. Shoitov
%T On repetitions of long tuples in a Markov chain
%J Diskretnaya Matematika
%D 2014
%P 79-89
%V 26
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DM_2014_26_3_a6/
%G ru
%F DM_2014_26_3_a6
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/

[1] Zubkov A. M., Mikhailov V. G., “Predelnye raspredeleniya sluchainykh velichin, svyazannykh s dlinnymi povtoreniyami v posledovatelnosti nezavisimykh ispytanii”, Teoriya veroyatn. i ee primen., 19:1 (1974), 173–181 | MR | Zbl

[2] Belyaev P. F., “O sovmestnom raspredelenii chastot dlinnykh $s$-tsepochek v multinomialnoi skheme s ravnoveroyatnymi iskhodami”, Teoriya veroyatn. i primen., 14:3 (1969), 540–546 | MR | Zbl

[3] Zubkov A. M., Mikhailov V. G., “O povtoreniyakh $s$-tsepochek v posledovatelnosti nezavisimykh velichin”, Teoriya veroyatn. i primen., 24:2 (1979), 267–279 | MR | Zbl

[4] Mikhailov V. G., Shoitov A. M., “Strukturnaya ekvivalentnost $s$-tsepochek v sluchainykh diskretnykh posledovatelnostyakh”, Diskretnaya matematika, 15:4 (2003), 7–34 | DOI | MR | Zbl

[5] Hansen N. R., “Local alignment of Markov chains”, Ann. Appl. Probab., 16:3 (2006), 1262–1296 | DOI | MR | Zbl

[6] Chryssaphinou O., Vaggelatou E., “Compound poisson approximation for multiple runs in a Markov chain”, Ann. Inst. Statist. Math., 54:2 (2002), 411–424 | DOI | MR | Zbl

[7] Belyaev P. F., “K voprosu o sovmestnom raspredelenii chastot $s$-tsepochek v slozhnykh tsepyakh Markova s bolshim chislom sostoyanii”, Teoriya veroyatn. i primen., 14:2 (1969), 333–339 | MR | Zbl

[8] Belyaev P. F., “O sovmestnom raspredelenii chastot iskhodov v tsepyakh Markova s bolshim chislom sostoyanii”, Teoriya veroyatn. i primen., 22:3 (1977), 534–545 | MR | Zbl

[9] Zubkov A. M., “Neravenstva dlya veroyatnostei perekhodov s zaprescheniyami i ikh primeneniya”, Matem. sb., 109(151):4(8) (1979), 491–532 | MR | Zbl

[10] Gantmakher F. R., Teoriya matrits, 5-e izd., FIZMATLIT, M., 2004, 576 pp. | MR

[11] Shoitov A. M., “Puassonovskoe priblizhenie dlya chisla povtorenii znachenii diskretnoi funktsii ot tsepochek”, Diskretnaya matematika, 17:2 (2005), 56–69 | DOI | MR | Zbl

[12] Barbour A. D., Holst L., Janson S., Poisson Approximation, Oxford University Press, Oxford, 1992, 277 pp. | MR | Zbl