Coupling and ergodic theorems for Markov chains with damping component

• Published in: Theory of probability and mathematical statistics 2019-01, Vol.101, p.243-264
• Main Authors: Silvestrov, D., Silvestrov, S., Abola, B., Biganda, P. S., Engström, C., Mango, J. M., Kakuba, G.
• Format: Article
• Language: English
• Subjects: Coupling; Damping component; Ergodic theorem; Information network; Markov chain; matematik; matematik/tillämpad matematik; Mathematics; Mathematics/Applied Mathematics; Rate of convergence; Regular perturba-tion; Regular perturbation; Research article; Singular perturbation; Triangular array mode
• ISSN: 0094-9000; 1547-7363; 1547-7363
• DOI: 10.1090/tpms/1124

Perturbed Markov chains are popular models for description of information networks. In such models, the transition matrix P0\mathbf {P}_0 of an information Markov chain is usually approximated by matrix Pε=(1−ε)P0+εD\mathbf {P}_{\varepsilon }=(1 - \varepsilon ) \mathbf {P}_0+\varepsilon \mathbf {D}, where D\mathbf {D} is a so-called damping stochastic matrix with identical rows and all positive elements, while ε∈[0,1]\varepsilon \in [0, 1] is a damping (perturbation) parameter. Using procedure of artificial regeneration for the perturbed Markov chain ηε,n\eta _{\varepsilon , n}, with the matrix of transition probabilities Pε\mathbf {P}_{\varepsilon }, and coupling methods, we get ergodic theorems, in the form of asymptotic relations for pε,ij(n)=Pi{ηε,n=j}\begin{equation*} p_{\varepsilon , ij}(n) =\mathsf {P}_i \{\eta _{\varepsilon , n}=j \} \end{equation*} as n→∞n \to \infty and ε→0\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 η0,n\eta _{0, n} split in several closed classes of communicative states and possibly a class of transient states, is investigated.