Geodesic
Joint distribution of the number of ones and the number of 1-runs in binary Markov sequences
Diskretnaya Matematika, Tome 16 (2004) no. 3, pp. 43-62.

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We describe the distributions of the number of ones, the number of runs of ones in a binary Markov sequence and their joint distribution. We find the generating functions of the distributions under consideration, calculate the means, variances and covariances. For these moments we give explicit and asymptotic formulas with estimates of accuracy. Formulas for the Gaussian approximations are also derived. We consider the corresponding operator equations. In this connection we describe a special model of random walks.This research was supported by the Russian Foundation for Basic Research, grant 02–01–0946. 
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L. Ja. Savel'ev; S. V. Balakin. Joint distribution of the number of ones and the number of 1-runs in binary Markov sequences. Diskretnaya Matematika, Tome 16 (2004) no. 3, pp. 43-62. http://geodesic.mathdoc.fr/item/DM_2004_16_3_a1/

[1] Markov A. A., Ischislenie veroyatnostei, Imperatorskaya Akademiya Nauk, Sankt-Peterburg, 1913

[2] Kolmogorov A. N., Teoriya veroyatnostei i matematicheskaya statistika, Nauka, Moskva, 1986 | MR

[3] Sarymsakov T. A., Osnovy teorii protsessov Markova, Gostekhizdat, Moskva, 1954 | MR

[4] Sirazhdinov S. Kh., Predelnye teoremy dlya tsepei Markova, Izd-vo AN UzSSR, Tashkent, 1955

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

[6] Mood A. M., “The distribution theory of runs”, Ann. Math. Stat., 11:4 (1940), 367–392 | DOI | MR | Zbl

[7] Hirano K., Aki S., “On the number of occurrences of success runs of length $k$ in a two-state Markov chain”, Statist. Sinica, 3 (1992), 313–320 | MR

[8] Uchida M., “On number of occurrences of success runs of specified length in a higher-order two-state Markov chain”, Ann. Inst. Statist. Math., 50:3 (1998), 587–601 | DOI | MR | Zbl

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

[10] Smirnov N. V., Sarmanov O. V., Zakharov V. K., “Lokalnaya predelnaya teorema dlya chisel perekhodov v tsepi Markova i ee primeneniya”, Dokl. AN SSSR, 167, no. 6, 1966, 1238–1241 | MR

[11] Sarmanov O. V., Zakharov V. K., “Kombinatornaya zadacha N. V. Smirnova”, Dokl. AN SSSR, 176, no. 3, 1967, 530–532 | MR | Zbl

[12] Zakharov V. K., Sarmanov O. V., “O zakone raspredeleniya chisla serii v odnorodnoi tsepi Markova”, Dokl. AN SSSR, 179, no. 3, 1968, 526–528

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

[14] Kolchin V. F., Sluchainye grafy, Fizmatlit, Moskva, 2000 | MR | Zbl

[15] Gamkrelidze N. G., Veroyatnost i matematicheskaya statistika, Entsiklopediya, Bolshaya Rossiiskaya entsiklopediya, Moskva, 1999 | MR

[16] Kolmogorov A. N., “Lokalnaya predelnaya teorema dlya klassicheskikh tsepei Markova”, Izvestiya AN SSSR. Ser. matem., 13 (1949), 281–300 | MR | Zbl

[17] Savelev L. Ya., “Serii v markovskikh posledovatelnostyakh”, Sib. matem. zh., 32:4 (1991), 116–132 | MR

[18] Savelev L. Ya., Markovskie matrichnye operatory, Dep. v VINITI 457-V 95, 1995