FORMULAE AND RECURSIONS FOR THE JOINT DISTRIBUTION OF SUCCESS RUNS OF SEVERAL LENGTHS

ANANT P. GODBOLE^{1∗}, STAVROS G. PAPASTAVRIDIS^{2} AND ROBERT S. WEISHAAR^{3∗}

1Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, U.S.A.

2Department of Mathematics, University of Athens, Panepistemiopolis, 15710 Athens, Greece 3Department of Mathematics, The Ohio State University, Columbus, OH 43210, U.S.A.

(Received November 15, 1993; revised March 18, 1996)

Abstract. Consider a sequence of n independent Bernoulli trials with the j-th trial having probability p_{j} of success, 1 ≤ j ≤ n. Let M(n, K ) and N(n, K ) denote, respectively, the r-dimensional random variables (M(n, k_{1}), …, M(n, k_{r})) and (N(n, k_{1}), …, N(n, k_{r})), where K = (k_{1}, k_{2}, …, k_{r}) and M(n, s) [N(n, s)] represents the number of overlapping [non-overlapping] success runs of length s. We obtain exact formulae and recursions for the probability distributions of M(n, K ) and N(n, K ). The techniques of proof employed include the inclusion-exclusion principle and generating function methodology. Our results have potential applications to statistical tests for randomness.

Key words and phrases: Overlapping and non-overlapping success runs, distributions of order k, generating functions, tests for randomness, inclusionexclusion.