The joint distribution of the number of ones and the number of 1-runs in binary Markov sequences

We describe the distributions of the number of ones, the number of runs of ones in a binary Markov sequence and their joint distribution. We find the generating functions of the distributions under consideration, calculate the means, variances and covariances. For these moments we give explicit and asymptotic formulas with estimates of accuracy. Formulas for the Gaussian approximations are also derived. We consider the corresponding operator equations. In this connection we describe a special model of random walks.