Distribution of runs in a sequence of exchangeable multi-state trials

Let {Xi}i≥1{Xi}i≥1 be a an infinite sequence of exchangeable multi-state trials with mm (m≥2m≥2) possible values. The exact distributions of runs, particularly the total number of runs, longest and shortest runs, in {Xi}i≥1{Xi}i≥1 are derived. Applications associated with multi-color Pólya urn sampling scheme and statistical process control are also provided.