Geodesic
Covering runs in binary Markov sequences
Diskretnaya Matematika, Tome 15 (2003) no. 1, pp. 50-76.

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

We describe distributions of the lengths of initial, covering, and final runs in binary Markov sequences. For the means and variances, we give exact and asymptotic formulas. We also give the generating functions. We observe that in Markov sequences the probabilities of run lengths do not necessarily decrease as the lengths grow, and hence, the corresponding distributions may be of quite complex form. We investigate conditions under which, due to the Markov property, the probabilities increase as the run lengths do. We consider operator equations which include final runs.This research was supported by the Russian Foundation for Basic Research, grant 02–01–00946. 
@article{DM_2003_15_1_a2,
     author = {L. Ja. Savel'ev and S. V. Balakin and B. V. Khromov},
     title = {Covering runs in binary {Markov} sequences},
     journal = {Diskretnaya Matematika},
     pages = {50--76},
     publisher = {mathdoc},
     volume = {15},
     number = {1},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2003_15_1_a2/}
}
TY  - JOUR
AU  - L. Ja. Savel'ev
AU  - S. V. Balakin
AU  - B. V. Khromov
TI  - Covering runs in binary Markov sequences
JO  - Diskretnaya Matematika
PY  - 2003
SP  - 50
EP  - 76
VL  - 15
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DM_2003_15_1_a2/
LA  - ru
ID  - DM_2003_15_1_a2
ER  - 
%0 Journal Article
%A L. Ja. Savel'ev
%A S. V. Balakin
%A B. V. Khromov
%T Covering runs in binary Markov sequences
%J Diskretnaya Matematika
%D 2003
%P 50-76
%V 15
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DM_2003_15_1_a2/
%G ru
%F DM_2003_15_1_a2
L. Ja. Savel'ev; S. V. Balakin; B. V. Khromov. Covering runs in binary Markov sequences. Diskretnaya Matematika, Tome 15 (2003) no. 1, pp. 50-76. http://geodesic.mathdoc.fr/item/DM_2003_15_1_a2/

[1] Markov A. A., Ischislenie veroyatnostei, Tipografiya Imperatorskoi Akademii Nauk, Sankt-Peterburg, 1913

[2] Romanovskii V. I., Diskretnye tsepi Markova, Gostekhizdat, Moskva, 1949

[3] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, t. 1, Mir, Moskva, 1984

[4] Savelev L. Ya., “Operatornye uravneniya dlya proizvodyaschikh funktsii”, Dokl. AN SSSR, 343:6 (1995), 740–742 | MR

[5] Savelyev L. Ya., “Runs in finite Markov chains”, Probabilistic Methods in Discrete Mathematics, VSP, Utrecht, 1993, 437–450 | MR

[6] Savelyev L. Ya., “Operator and recursion equations for runs in random sequences”, Probabilistic Methods in Discrete Mathematics, VSP, Utrecht, 1997, 65–86 | MR | Zbl

[7] Savelev L. Ya., “Sluchainye sootvetstviya, dvoichnye matritsy i serii”, Diskretnaya matematika, 11:4 (1999), 3–26 | MR

[8] Kostochka A. V., Mazurov V. D., Savelev L. Ya., “Chislo $q$-ichnykh slov s ogranicheniyami na dlinu maksimalnoi serii”, Diskretnaya matematika, 10:1 (1998), 10–19 | Zbl

[9] Savelev L. Ya., “Maksimum dlin serii v obobschennykh posledovatelnostyakh Bernulli”, Diskretnaya matematika, 11:1 (1999), 29–52 | MR

[10] Uitteker E. T., Vatson Dzh. N., Kurs sovremennogo analiza, Gostekhizdat, Moskva, 1962

[11] Savelev L. Ya., Markovskie matrichnye operatory, Dep. v VINITI 457-V95, 1995

[12] Fu J. C., Koutras M. V., “Distribution theory of runs: A Markov chain approach”, J. Amer. Statist. Assoc., 89 (1994), 1050–1058 | DOI | MR | Zbl

[13] Koutras M. V., Alexandrou V. A., “Runs, scans and urn model distributions: A unified Markov chain approach”, Ann. Inst. Statist. Math., 47 (1995), 519–529 | DOI | MR

[14] Balakrishnan N., “Joint distributions of numbers of success-runs and failures until the first consecutive $k$ successes in a binary sequence”, Ann. Inst. Statist. Math., 49 (1997), 519–529 | DOI | MR | Zbl

[15] Uchida M., “On number of occurrences of success runs of specified length in a higher-order two-state Markov chain”, Ann. Inst. Statist. Math., 50 (1998), 587–601 | DOI | MR | Zbl

[16] Aki S., Hirano K., “Numbers of success-runs of specified length until certain stopping time rules and generalized binomial distributions of order $k$”, Ann. Inst. Statist. Math., 52 (2000), 767–777 | DOI | MR | Zbl