On the central limit theorem for homogeneous discrete-time nonlinear Markov chains

The class of nonlinear Markov processes is characterized by the dependence of the current state of the process on its current distribution in addition to the dependence on the previous state. Due to this feature, these processes are characterized by complex limit behavior and ergodic properties, for which the usual criteria for Markov processes are not sufficient. Being a subclass of nonlinear Markov processes, nonlinear Markov chains have inherited these features. In this paper, conditions for the fulfillment of the central limit theorem for homogeneous nonlinear Markov chains in discrete time and with a discrete finite state space are studied. Also, a brief review of known results on the ergodic properties of nonlinear Markov chains is included. The obtained result complements the existing results in this area and may be useful for further applications in statistics.