Asymptotic periodicity of the variances and covariances of the state sizes in non-homogeneous Markov systems

In this paper the authors study the asymptotic periodicity of the sequence of means, variances and covariances of the state sizes of non-homogeneous Markov systems. It is proved that under the assumption that the sequence of the extended stochastic transition matrices converge to a matrix which is an irreducible stochastic matrix of period d, and all the matrices in this sequence have the same incidence matrix, then the sequence of means, variances and covariances splits into d subsequences which converge. Finally, they discuss the application of the present results in a manpower system.