Abstract:
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.
Citation:
L. Ja. Savel'ev, S. V. Balakin, “Joint distribution of the number of ones and the number of 1-runs in binary Markov sequences”, Diskr. Mat., 16:3 (2004), 43–62; Discrete Math. Appl., 14:4 (2004), 353–372
\Bibitem{SavBal04}
\by L.~Ja.~Savel'ev, S.~V.~Balakin
\paper Joint distribution of the number of ones and the number of 1-runs in binary Markov sequences
\jour Diskr. Mat.
\yr 2004
\vol 16
\issue 3
\pages 43--62
\mathnet{http://mi.mathnet.ru/dm162}
\crossref{https://doi.org/10.4213/dm162}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2109796}
\zmath{https://zbmath.org/?q=an:1103.60302}
\transl
\jour Discrete Math. Appl.
\yr 2004
\vol 14
\issue 4
\pages 353--372
\crossref{https://doi.org/10.1515/1569392041938802}
Linking options:
https://www.mathnet.ru/eng/dm162
https://doi.org/10.4213/dm162
https://www.mathnet.ru/eng/dm/v16/i3/p43
This publication is cited in the following 10 articles:
N. M. Mezhennaya, “Otsenka dlya raspredeleniya chisel serii v sluchainoi posledovatelnosti, upravlyaemoi statsionarnoi tsepyu Markova”, PDM, 2017, no. 35, 14–28
N. M. Mezhennaya, “O predelnom raspredelenii chisla serii v polinomialnoi posledovatelnosti, upravlyaemoi tsepyu Markova”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 26:3 (2016), 324–335
Corzo J.A., Vergara M.E., “A Data Driven Runs Test to Identify First Order Positive Markovian Dependence”, Hacet. J. Math. Stat., 45:2 (2016), 521–540
L. Ya. Savelev, S. V. Balakin, “Nekotorye primeneniya stokhasticheskoi teorii serii”, Sib. zhurn. industr. matem., 15:3 (2012), 111–123
L. Ya. Savelev, S. V. Balakin, “A stochastic model of a digit transfer by computing”, Num. Anal. Appl., 6:1 (2013), 71–76
V. A. Barvinok, V. I. Bogdanovich, A. N. Plotnikov, “K voprosu o predelnom raspredelenii serii v sluchainoi dvoichnoi posledovatelnosti”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 4(29) (2012), 56–71
Kargapolova N.A., Ogorodnikov V.A., “Inhomogeneous Markov Chains with Periodic Matrices of Transition Probabilities and their Application to Simulation of Meteorological Processes”, Russ. J. Numer. Anal. Math. Model, 27:3 (2012), 213–228
V. S. Lugavov, “On some class of functionals on transitions of Markov chain”, J. Math. Sci., 198:5 (2014), 580–601
L. Ya. Savelyev, S. V. Balakin, “A combinatorial approach to calculation of moments of characteristics of runs in ternary Markov sequences”, Discrete Math. Appl., 21:1 (2011), 47–67
M. M. Lavrent'ev, L. Ja. Savel'ev, S. V. Balakin, “Special operator equations”, J. Appl. Industr. Math., 3:2 (2009), 246–258