# The *m*th longest runs of multivariate random sequences

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## Abstract

The distributions of the *m*th longest runs of multivariate random sequences are considered. For random sequences made up of *k* kinds of letters, the lengths of the runs are sorted in two ways to give two definitions of run length ordering. In one definition, the lengths of the runs are sorted separately for each letter type. In the second definition, the lengths of all the runs are sorted together. Exact formulas are developed for the distributions of the *m*th longest runs for both definitions. The derivations are based on a two-step method that is applicable to various other runs-related distributions, such as joint distributions of several letter types and multiple run lengths of a single letter type.

## Keywords

Generating function Combinatorial identities Randomness test Distribution-free statistical test Runs length test Biological sequence analysis## 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.

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