Exact and approximate runs distributions

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Let R = Rn denote the total (and unconditional) number of runs of successes or failures in a sequence of n Bernoulli (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 - 1/2. This yields an unconditional version of the classical result of Wald and Wolfowitz. © 1992, Taylor & Francis Group, LLC. All rights reserved.

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Communications in Statistics - Theory and Methods