On Sooner and Later Problems Between Success and Failure Runs
Sigeo Aki · Birkhäuser Boston eBooks · 1997
Let X 0, X 1, X 2,… be a sequence of {0, 1}-valued Markov chain. Let E 1 denote a run of “1” of length k and E 0 denote a run of “0” of length r. We observe which run comes sooner or later in the sequence X 0, X 1, X 2, …. The exact distributions of the numbers of overlapping sooner runs and nonoverlapping sooner runs until the later run occurs (for the first time) are derived. Let F 1 be a success run of length k or more and let F 0 be a failure run of length r or more. The exact distribution of the number of occurrences of the sooner event until the first occurrence of the later event between F 1 and F 0 is also studied. Further, when X’s have more than two values, more general problems are discussed and the exact joint distribution of the numbers of occurrences of the first, the second,…, and the j-th runs until the j-th run occurs (for the first time) is obtained in the case of independent trials.