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Distribution of the Number of Successes in Success Runs of Length at Least k in Higher-Order Markovian Sequences

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Abstract

We consider the distribution of the number of successes in success runs of length at least k in a binary sequence. One important application of this statistic is in the detection of tandem repeats among DNA sequence segments. In the literature, its distribution has been computed for independent sequences and Markovian sequences of order one. We extend these results to Markovian sequences of a general order. We also show that the statistic can be represented as a function of the number of overlapping success runs of lengths k and k + 1 in the sequence, and give immediate consequences of this representation.

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References

  • G. Benson, “Tandem repeats finder: A program to analyze DNA sequences,” Nucleic Acids Research vol. 27 pp. 573–580, 1999.

    Google Scholar 

  • G. Benson and X. Su, “On the distribution of k-tuple matches for sequence homology: A constant time exact calculation of the variance,” Journal of Computational Biology vol. 5 pp. 87–100, 1998.

    Article  Google Scholar 

  • J. C. Fu and M. V. Koutras, “Distribution theory of runs: A Markov chain approach,” Journal of the American Statistical Association vol. 89 pp. 1050–1058, 1994.

    MathSciNet  MATH  Google Scholar 

  • J. C. Fu, W. Y. W. Lou, Z.-D. Bai, and G. Li, “The exact and limiting distributions for the number of successes in success runs within a sequence of Markov-dependent two-state trials,” Annals of the Institute of Statistical Mathematics vol. 54(4) pp. 719–730, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  • W. Y. W. Lou, “The exact distribution of the k-tuple statistic for sequence homology,” Statistics & Probabability Letters vol. 61 pp. 51–59, 2003.

    MATH  MathSciNet  Google Scholar 

  • D. E. K. Martin, “On the distribution of the number of successes in fourth- or lower-order Markovian trials,” Computers & Operations Research vol. 27(2) pp. 93–109, 2000.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Howard University, Washington, DC, 20059, USA

    Donald E. K. Martin

  2. Statistical Research Division, US Bureau of the Census, Washington, DC, 20233-9100, USA

    Donald E. K. Martin

Authors
  1. Donald E. K. Martin
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Correspondence to Donald E. K. Martin.

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AMS 2000 Subject Classification

60E05, 60J05

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Martin, D.E.K. Distribution of the Number of Successes in Success Runs of Length at Least k in Higher-Order Markovian Sequences. Methodol Comput Appl Probab 7, 543–554 (2005). https://doi.org/10.1007/s11009-005-5007-9

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  • DOI: https://doi.org/10.1007/s11009-005-5007-9

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