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

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