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Formulae and recursions for the joint distributions of success runs of several lengths in a two-state Markov chain. (English) Zbl 0938.60063

Denote by \(Y_i\) the number of success runs of length \(k_i\) \((i= 1,\dots, r)\) in a time-homogeneous \(\{0,1\}\)-valued Markov chain \(X_1,X_2,\dots, X_n\), and set \(Y= (Y_1,Y_2,\dots, Y_r)\) and \(k= (k_1,\dots, k_r)\). The authors consider the four most important types of runs (overlapping runs, non-overlapping runs, runs with a specified length or more, and runs with a specified length) and derives exact and recurrence formulas for the probability function and probability generating function of \(Y\) for each one of them.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60E10 Characteristic functions; other transforms
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References:

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