(Received February 1, 1996; revised July 1, 1996)
Abstract. Joint distributions of the numbers of failures, successes and success-runs of length less than k until the first consecutive k successes in a binary sequence were derived recently by Aki and Hirano (1995, Ann.Inst. Statist. Math., 47, 225-235). In this paper, we present an alternate derivation of these results and also use this approach to establish some additional results. Extensions of these results to binary sequences of order h are also presented.
Key words and phrases: Probability generating function, waiting time, binary sequence of order k, geometric distribution of order k.