  A new approach is developed for evaluating the convergence rate for nonlinear
Markov chains (MC) based on the recently developed spectral radius technique of
markovian coupling for linear MC and the idea of small nonlinear perturbations
of linear MC. The method further enhances recent advances in the problem of
convergence for such models. The new convergence rate may be used, in
particular, for the justification of $D$-condition in the Extreme Values
theory.
