Distributions of Numbers of Success-Runs Until the First Consecutive k Successes in Higher Order Markov Dependent Trials
Katuomi Hirano, Sigeo Aki, Masayuki Uchida · Birkhäuser Boston eBooks · 1997
The distributions of numbers of overlapping and non-overlapping occurrences of success-runs of length l , and the distributions of numbers of occurrences of success-runs of exact length l and of length l or more until the first occurrence of success-run of length k in the m -th order Markov dependent trials are studied. When m ≤ l < k the derived distributions do not depend on the initial distribution of the m -th order Markov chain and they are shown to be equal to the corresponding distributions considered in independent trials with a success probability whose value is given by the transition probability that a success occurs after a success-run of length m in the m -th order Markov chain. When l < m the distributions depend on the initial distribution of the m -th order Markov chain and are not necessarily so simple as the above case. A method for deriving the probability generating function of the conditional distribution of the number of overlapping occurrences of success-runs of length l until the first occurrence of success-run of length k under each initial condition is given.