Abstract: In this paper, the limit properties of second order nonhomogeneous Markov information source are studied by using the analytic approach, and a limit theorem for the averages of the functions of three variables of this information source, including several limit properties which all second order nonhomogeneous Markov information source possess and several limit properties of the relative entropy densities of this information source, are obtained. We also give an extension of Shannon theorem to the case of second order nonhomogeneous Markov information source.