On the distribution of the total number of run lengths

In the present paper, we study the distribution of a statistic utilizing the runs length of "reasonably long" series of alike elements (success runs) in a sequence of binary trials. More specifically, we are looking at the sum of exact lengths of subsequences (strings) consisting of k or more consecutive successes (k is a given positive integer). The investigation of the statistic of interest is accomplished by exploiting an appropriate generalization of the Markov chain embedding technique introduced by Fu and Koutras (1994, J. Amer. Statist. Assoc., 89, 1050-1058) and Koutras and Alexandrou (1995, Ann. Inst. Statist. Math., 47, 743-766). In addition, we explore the conditional distribution of the same statistic, given the number of successes and establish statistical tests for the detection of the null hypothesis of randomness versus the alternative hypothesis of systematic clustering of successes in a sequence of binary outcomes.