Number of appearances of events in random sequences: a new generating function approach to Type II and Type III runs

  • Yong Kong


Distributions of runs of length at least k (Type II runs) and overlapping runs of length k (Type III runs) are derived in a unified way using a new generating function approach. A new and more compact formula is obtained for the probability mass function of the Type III runs.


Runs statistics Generating function Asymptotic distributions Factorial moments Wilf-Zeilberger method  



This work was supported in part by the Clinical and Translational Science Award UL1 RR024139 from the National Center for Research Resources, National Institutes of Health.

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  1. Balakrishnan, N., Koutras, M. V. (2002). Runs and scans with applications. New York, NY: Wiley.Google Scholar
  2. Flajolet, P., Sedgewick, R. (2009). Analytic combinatorics. New York, NY: Cambridge University Press.Google Scholar
  3. Fu, J. C., Koutras, M. V. (1994). Distribution theory of runs: A Markov chain approach. Journal of the American Statistical Association, 89, 1050–1058.Google Scholar
  4. Hirano, K., Aki, S. (1993). On number of occurrences of success runs of specified length in a two-state Markov chain. Statistica Sinica, 3, 313–320.Google Scholar
  5. Kong, Y. (2006). Distribution of runs and longest runs: A new generating function approach. Journal of the American Statistical Association, 101, 1253–1263.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Kong, Y. (2014). Number of appearances of events in random sequences: A new approach to non-overlapping runs. Communications in Statistics-Theory and Methods (To appear).Google Scholar
  7. Kong, Y. (2015). Distributions of runs revisited. Communications in Statistics-Theory and Methods, 44, 4663–4678.Google Scholar
  8. Koutras, M. (1997). Waiting times and number of appearances of events in a sequence of discrete random variables. In N. Balakrishnan (Ed.), Advances in combinatorial methods and applications to probability and statistics (pp. 363–384). Boston, MA: Birkhäuser.CrossRefGoogle Scholar
  9. Koutras, M. V., Alexandrou, V. A. (1995). Runs, scans and urn model distributions: A unified Markov chain approach. Annals of the Institute of Statistical Mathematics, 47(4), 743–766.Google Scholar
  10. Mood, A. M. (1940). The distribution theory of runs. Annals of Mathematical Statistics, 11, 367–392.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Petkovsěk, M., Wilf, H. S., Zeilberger, D. (1996). A = B. Wellesley, MA: A K Peters Ltd.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2015

Authors and Affiliations

  1. 1.Department of Molecular Biophysics and Biochemistry, W.M. Keck Foundation Biotechnology Resource LaboratoryYale UniversityNew HavenUSA

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