Exact and approximate runs distributions

Let R = R_n denote the total (and unconditional) number of runs of successes or failures in a sequence of n Bernoulll (p) trials, where p is assumed to be known throughout. The exact distribution of R is related to a convolution of two negative binomial random variables with parameters p and q (=1-p). Using the representation of R as the sum of 1 - dependent indicators, a Berry - Esséen theorem is derived; the obtained rate of sup norm convergence is O(n^-½). This yields an unconditional version of the classical result of Wald and Wolfowitz (1940). The Stein - Chen method for m - dependent random variables is used, together with a suitable coupling, to prove a Poisson limit theorem for R. but with the limiting support set being the set of odd integers, Total variation error bounds (of order O(p) are found for the last result. Applications are indicated.