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

Abstract

Consider a sequence of n Bernoulli (Success–Failure or 1–0) trials. The exact and limiting distribution of the random variable E n , k denoting the number of success runs of a fixed length k , 1 ≤ 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 E n , k are obtained using Poisson and normal approximations. The exact mean and variance of E n , k which are given in explicit forms are also used to derive bounds and an additional approximation of the distribution of E n , k . Numbers, associated with E n , 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.