On success runs of a fixed length in Bernoulli sequences: Exact and asymptotic results

Consider a sequence of nn Bernoulli (Success–Failure or 1–0) trials. The exact and limiting distribution of the random variable En,kEn,k denoting the number of success runs of a fixed length kk, 1≤k≤n1≤k≤n, is derived along with its mean and variance. An associated waiting time is examined as well. The exact distribution is given in terms of binomial coefficients and an extension of it covering exchangeable sequences is also discussed. Limiting distributions of En,kEn,k are obtained using Poisson and normal approximations. The exact mean and variance of En,kEn,k which are given in explicit forms are also used to derive bounds and an additional approximation of the distribution of En,kEn,k. Numbers, associated with En,kEn,k and related random variables, counting binary strings and runs of 1’s useful in applications of computer science are provided. The overall study is illustrated by an extensive numerical experimentation.