Coupling and ergodic theorems for Markov chains with damping component

Perturbed Markov chains are popular models for description of information networks. In such models, the transition matrix $\mathbf {P}_0$ of an information Markov chain is usually approximated by matrix $\mathbf {P}_{\varepsilon }=(1 - \varepsilon ) \mathbf {P}_0+\varepsilon \mathbf {D}$, where $\mathbf {D}$ is a so-called damping stochastic matrix with identical rows and all positive elements, while $\varepsilon \in [0, 1]$ is a damping (perturbation) parameter. Using procedure of artificial regeneration for the perturbed Markov chain $\eta _{\varepsilon , n}$, with the matrix of transition probabilities $\mathbf {P}_{\varepsilon }$, and coupling methods, we get ergodic theorems, in the form of asymptotic relations for \begin{equation*} p_{\varepsilon , ij}(n) =\mathsf {P}_i \{\eta _{\varepsilon , n}=j \} \end{equation*} as $n \to \infty$ and $\varepsilon \to 0$, and explicit upper bounds for the rates of convergence in such theorems. In particular, the most difficult case of the model with singular perturbations, where the phase space of the unperturbed Markov chain $\eta _{0, n}$ split in several closed classes of communicative states and possibly a class of transient states, is investigated.