The exact and asymptotic distribution of overlapping success runs

Let M = M n,k and N = N n,k denote, respectively, the number of overlapping and non-overlapping success runs of fixed length k in n Bernoulli (p) trials. We derive an alternative formula for P(M = x) which is substantially simpler to apply than the one in Ling (1988). Our method of proof brings out several connections between the distributions of M and N. Poisson approximations and a Berry-Esséen theorem are obtained for M, and applications are indicated.