  The paper studies an improved estimate for the rate of convergence for
nonlinear homogeneous discrete-time Markov chains. These processes are
nonlinear in terms of the distribution law. Hence, the transition kernels are
dependent on the current probability distributions of the process apart from
being dependent on the current state. Such processes often act as limits for
large-scale systems of dependent Markov chains with interaction. The paper
generalizes the convergence results by taking the estimate over two steps. Such
an approach keeps the existence and uniqueness results under assumptions that
are analogical to the one-step result. It is shown that such an approach may
lead to a better rate of convergence. Several examples provided illustrating
the fact that the suggested estimate may have a better rate of convergence than
the original one. Also, it is shown that the new estimate may even be
applicable in some cases when the conditions of the result on one step cannot
guarantee any convergence. Finally, these examples depict that the original
conditions may not be an obstacle for the convergence of nonlinear Markov
chains.
