Formulae and recursions for the joint distribution of success runs of several lengths

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Consider a sequence of n independent Bernoulli trials with the j-th trial having probability Pj of success, 1 ≤ j ≤ n. Let M(n, K) and N(n,K) denote, respectively, the r-dimensional random variables (M(n, k1), ..., M(n,kr)) and (N (n, k1),..., N(n, kr)), where K = (k1, k2,..., kr) 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.

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Annals of the Institute of Statistical Mathematics