Geodesic
On a question of limiting distribution of series in~random binary sequence
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 4 (2012), pp. 56-71.

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Limiting forms of distribution of length of the maximum series of successes in random binary sequences, formed in Bernulli–Markov's chain and in Polya's scheme which is an equivalent to local trends of a time series of strictly stationary process are investigated. More simple and added proof of theorems of the law of the big numbers for series of both types is offered. For series of the second type the effect of the cyclic bimorphism of the limiting law with degeneration on one of the phases and the convergence according to the probability on set no more, than four consequent values of the natural series is established.
Keywords: random sequence, the maximum series of successes, limiting theorems, convergence on probability.
Mots-clés : convergence on distribution 
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V. A. Barvinok; V. I. Bogdanovich; A. N. Plotnikov. On a question of limiting distribution of series in~random binary sequence. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 4 (2012), pp. 56-71. http://geodesic.mathdoc.fr/item/VSGTU_2012_4_a5/

[1] Samarova S. S., “On the length of the longest head-run for a Markov chain with two states”, Theory Probab. Appl., 26:3 (1982), 498–509 | DOI | MR | Zbl

[2] Uspenskiy V. A., Semenov A. L., Shen' A. Kh., “Can an individual sequence of zeros and ones be random?”, Russian Math. Surveys, 45:1 (1990), 121–189 | DOI | MR | Zbl

[3] V'yugin V. V., “On the longest head-run in an individual random sequence”, Theory Probab. Appl., 42:3 (1998), 541–546 | DOI | DOI | MR | Zbl

[4] Savelyev L. Ya., Balakin S. V., “The joint distribution of the number of ones and the number of 1-runs in binary Markov sequences”, Discrete Math. Appl., 14:4 (2004), 353–372 | DOI | DOI | MR | Zbl

[5] An introduction to probability theory and its applications, v. 1, Wiley Sons, New York, 1968, 509 pp.; Feller V., Vvedenie v teoriyu veroyatnostei i eë prilozheniya, v. 1, Mir, M., 1984, 499 pp.

[6] Knuth D., The Art of Computer Programming, v. 2, Seminumerical Algorithms, Addison-Wesley, Reading, Massachusetts, 1997, xiv+762 pp. ; Knut D., Iskusstvo programmirovaniya, v. 2, Poluchislennye metody, Vilyams, M., 2007, 832 pp. | MR

[7] Plotnikov A. N., “On the paradox of the law of large numbers for the maximum series in sequential sampling”, Izvestiya Samarskogo nauchnogo tsentra RAN, 11:5 (2009), 122–126

[8] Plotnikov A. N., “Law distribution of length for maximal series and its statistical applications”, Izvestiya Samarskogo nauchnogo tsentra RAN, 8:4 (2006), 1047–1056 | MR

[9] Barvinok V. A., Bogdanovich V. I., Plotnikov A. N., “On the spectral structure of the series in sequential sampling for stationary process”, Obozr. Prikl. Promyshl. Matem., 17:3 (2010), 382–384