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

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## Abstract

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.

### Keywords

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

### Acknowledgments

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.

## Supplementary material

10463_2015_549_MOESM1_ESM.mw (51 kb)

### References

- Balakrishnan, N., Koutras, M. V. (2002).
*Runs and scans with applications*. New York, NY: Wiley.Google Scholar - Flajolet, P., Sedgewick, R. (2009).
*Analytic combinatorics*. New York, NY: Cambridge University Press.Google Scholar - 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 - 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 - Kong, Y. (2006). Distribution of runs and longest runs: A new generating function approach.
*Journal of the American Statistical Association*,*101*, 1253–1263.MathSciNetCrossRefMATHGoogle Scholar - 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 - Kong, Y. (2015). Distributions of runs revisited.
*Communications in Statistics-Theory and Methods*,*44*, 4663–4678.Google Scholar - 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 - 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 - Mood, A. M. (1940). The distribution theory of runs.
*Annals of Mathematical Statistics*,*11*, 367–392.MathSciNetCrossRefMATHGoogle Scholar - 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