# 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)

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## Copyright information

© The Institute of Statistical Mathematics, Tokyo 2015