Run statistics in a sequence of arbitrarily dependent binary trials

Abstract

Let {Z (i) } (ia parts per thousand yen1) be an arbitrary sequence of trials with two possible outcomes either success (1) or failure (0). General expressions for the exact distributions of runs, both success and failure, in Z (1), . . . , Z (n) are presented. Our method is based on the use of joint distribution of success and failure run lengths and unifies the results on distribution of runs. As a special case of our results we obtain the distributions of runs for various binary sequences. As illustrated in the paper the results enable us to derive the distribution of runs for binary trials arising in urn models.