Counting certain binary strings. (English) Zbl 1232.62071

Summary: Consider a sequence of exchangeable or independent binary (i.e., zero-one) random variables. Numbers of strings with a fixed number of ones between two subsequent zeros are studied under an overlapping enumeration scheme. The respective waiting times are examined as well. Exact probability functions are obtained by means of combinatorial analysis and via recursive schemes in the case of an exchangeable and of an independent sequence, respectively. Explicit formulae for the mean values and variances of the number of strings are provided for both types of the sequences. For a Bernoulli sequence the asymptotic normality of the numbers of strings is established too. Indicative exchangeable and independent sequences, combined with numerical examples, clarify further the theoretical results.


62G10 Nonparametric hypothesis testing
60G09 Exchangeability for stochastic processes
62E15 Exact distribution theory in statistics
65C60 Computational problems in statistics (MSC2010)
Full Text: DOI


[1] Antzoulakos, D.L., On waiting time for patterns in a sequence of multistate trials, Journal of applied probability, 38, 508-518, (2001) · Zbl 0985.60072
[2] Antzoulakos, D.L.; Bersimis, S.; Koutras, M.V., On the distribution of the total number of run lengths, Annals of the institute of statistical mathematics, 55, 865-884, (2003) · Zbl 1047.62013
[3] Balakrishnan, N.; Koutras, M.V., Runs and scans with applications, (2002), Wiley New York · Zbl 0991.62087
[4] Blom, G.; Holst, L.; Sandell, D., Problems and snapshots from the world of probability, (1994), Springer-Verlag New York · Zbl 0785.60001
[5] Boutsikas, M.V.; Koutras, M.V., Modeling claim exceedances over thresholds, Insurance: mathematics and economics, 30, 67-83, (2002) · Zbl 1055.91040
[6] Charalambides, C.A., Enumerative combinatorics, (2002), Chapman and Hall/CRC Boca Raton · Zbl 1001.05001
[7] Chern, H.-H.; Hwang, H.-K., Limit distribution of the number of consecutive records, Random structures and algorithm, 26, 404-417, (2005) · Zbl 1076.62014
[8] Chern, H.-H.; Hwang, H.-K.; Yeh, Y.-N., Distribution of the number of consecutive records, Random structures and algorithm, 17, 169-196, (2000) · Zbl 0969.60017
[9] Dafnis, S.D.; Philippou, A.N., On waiting time distributions for the occurrence of patterns, (), 1697-1704
[10] Dafnis, S.D.; Antzoulakos, D.L.; Philippou, A.N., Distributions related to \((k_1, k_2)\) events, Journal of statistical planning and inference, 140, 1691-1700, (2010) · Zbl 1195.60024
[11] Dafnis, S.D., Philippou, A.N., Antzoulakos, D.L., 2010b. Distributions of patterns of two successes separated by a string of k−2 failures. Statistical Papers, doi:10.1007/s00362-010-0340-7, pp. 1-22. · Zbl 1440.60018
[12] Demir, S.; Eryilmaz, S., Run statistics in a sequence of arbitrarily dependent binary trials, Statistical papers, 51, 959-973, (2010) · Zbl 1247.62037
[13] Eryilmaz, S., Discrete scan statistics generated by exchangeable binary trials, Journal of applied probability, 47, 1084-1092, (2010) · Zbl 1205.60032
[14] Eryilmaz, S., Joint distribution of run statistics in partially exchangeable processes, Statistics & probability letters, 81, 163-168, (2011) · Zbl 1205.60033
[15] Feller, W., 1968. An Introduction to Probability Theory and Its Applications, vol. 1, third ed. Wiley, New York. · Zbl 0155.23101
[16] Fu, J.C.; Lou, W.Y., Distribution theory of runs and patterns and its applications: A finite Markov chain imbedding approach, (2003), World Scientific New Jersey · Zbl 1030.60063
[17] George, E.O.; Bowman, D., A full likelihood procedure for analyzing exchangeable binary data, Biometrics, 51, 512-523, (1995) · Zbl 0832.62092
[18] Glaz, J.; Balakrishnan, N., Scan statistics and applications, (1999), Birkhauser Boston · Zbl 0919.00015
[19] Hirano, K.; Aki, S.; Kashiwagi, N.; Kuboki, H., On Ling’s binomial and negative binomial distributions of order k, Statistics & probability letters, 11, 503-509, (1991) · Zbl 0728.60017
[20] Hoeffding, W.; Robbins, H., The central limit theorem for dependent random variables, Duke mathematical journal, 15, 773-780, (1948) · Zbl 0031.36701
[21] Holst, L., Counts of failure strings in certain Bernoulli sequences, Journal of applied probability, 44, 824-830, (2007) · Zbl 1132.60011
[22] Holst, L., The number of two consecutive successes in a hope-polya urn, Journal of applied probability, 45, 901-906, (2008) · Zbl 1151.60002
[23] Holst, L., A note on embedding certain Bernoulli sequences in marked Poisson processes, Journal of applied probability, 45, 1181-1185, (2008) · Zbl 1154.60009
[24] Holst, L., On consecutive records in certain Bernoulli sequences, Journal of applied probability, 46, 1201-1208, (2009) · Zbl 1187.60006
[25] Holst, L., 2010. A note on records in a random sequence. Arkiv för Matematik, doi:10.1007/s11512-010-0131-3, pp. 1-6.
[26] Inoue, K.; Aki, S.; Hirano, K., Distributions of simple patterns in some kinds of exchangeable sequences, Journal of statistical planning and inference, 141, 2532-2544, (2011) · Zbl 1213.62022
[27] Johnson, N.; Kotz, S., Urn models and their applications, (1977), John Wiley New York
[28] Koutras, M.V., On the waiting time distribution in a sequence of Bernoulli trials, Annals of the institute of statistical mathematics, 48, 789-806, (1996) · Zbl 1002.60517
[29] Koutras, M.V.; Alexandrou, V.A., Runs, scans and urn model distributions: a unified Markov chain approach, Annals of the institute of statistical mathematics, 47, 743-766, (1995) · Zbl 0848.60021
[30] Koutras, M.V.; Alexandrou, V.A., Non-parametric randomness tests based on success runs of fixed length, Statistics & probability letters, 32, 393-404, (1997) · Zbl 0901.62063
[31] Ling, K.D., On binomial distributions of order k, Statistics & probability letters, 6, 247-250, (1988) · Zbl 0638.60024
[32] Mahmoud, H.M., Polya urn models, (2009), CRC Press Boca Raton · Zbl 1149.60005
[33] Makri, F.S., On occurrences of F-S strings in linearly and circularly ordered binary sequences, Journal of applied probability, 47, 157-178, (2010) · Zbl 1191.60018
[34] Makri, F.S.; Philippou, A.N.; Psillakis, Z.M., Success run statistics defined on an urn model, Advances in applied probability, 39, 991-1019, (2007) · Zbl 1135.60009
[35] Makri, F.S.; Psillakis, Z.M., Bounds for reliability of k-within connected-(r,s)-out-of-(m,n) failure systems, Microelectronics and reliability, 37, 1217-1224, (1997)
[36] Makri, F.S., Psillakis, Z.M., 2009. On runs of length exceeding a threshold: normal approximation. Statistical Papers, doi:10.1007/s00362-009-0268-y, pp. 1-21. · Zbl 1230.62017
[37] Makri, F.S.; Psillakis, Z.M., On success runs of length exceeded a threshold, Methodology and computing in applied probability, 13, 269-305, (2011) · Zbl 1230.60012
[38] Makri, F.S.; Psillakis, Z.M., On success runs of a fixed length in Bernoulli sequences: exact and asymptotic results, Computers & mathematics with applications, 61, 761-772, (2011) · Zbl 1217.60010
[39] Mood, A.M., The distribution theory of runs, Annals of mathematical statistics, 11, 367-392, (1940) · Zbl 0024.05301
[40] Mori, T.F., On the distributions of sums of overlapping products, Acta scientiarum mathematicarum (Szeged), 67, 833-841, (2001) · Zbl 1001.60015
[41] Nevzorov, V.B., Records: mathematical theory, (2001), American Mathematical Society Providence · Zbl 0989.62029
[42] Roussas, G.G., Introduction to probability, (2007), Academic Press San Diego · Zbl 1113.60001
[43] Sarkar, A.; Sen, K.; Anuradha, Waiting time distribution of runs in higher order Markov chains, Annals of the institute of statistical mathematics, 56, 317-349, (2004) · Zbl 1077.60052
[44] Sen, K.; Goyal, B., Distributions of patterns of two failures separated by success runs of length k, Journal of the Korean statistical society, 33, 35-58, (2004)
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