On the rate of convergence of Borovkov's multidimensional ergodic Markov chain

This paper is concerned with the multidimensional Markov chain {X(n)}={X(x,n)}, considered by Borovkov, with the form X(n+1)=X(n)+ξ(X(n),n), where the distribution of ξ(x,n)=dξ(x) depends only on x. Sufficient conditions on moments of |ξ(x)| are established for the Markov chain {X(n)} to have rate of convergence results (i.e. geometric ergodicity or sub-geometric rates for a variety of rate functions).