Iterative aggregation/disaggregation techniques for nearly uncoupled markov chains

TLDR: It is shown that the method of Takahashi corresponds to a modified block Gauss-Seidel step and aggregation, whereas that of Vantilborgh corresponds to an modified block Jacobistep and aggregation.

Abstract: Iterative aggregation/disaggregation methods provide an efficient approach for computing the stationary probability vector of nearly uncoupled (also known as nearly completely decomposable) Markov chains. Three such methods that have appeared in the literature recently are considered and their similarities and differences are outlined. Specifically, it is shown that the method of Takahashi corresponds to a modified block Gauss-Seidel step and aggregation, whereas that of Vantilborgh corresponds to a modified block Jacobi step and aggregation. The third method, that of Koury et al., is equivalent to a standard block Gauss-Seidel step and iteration. For each of these methods, a lemma is established, which shows that the unique fixed point of the iterative scheme is the left eigenvector corresponding to the dominant unit eigenvalue of the stochastic transition probability matrix. In addition, conditions are established for the convergence of the first two of these methods; convergence conditions for the third having already been established by Stewart et al. All three methods are shown to have the same asymptotic rate of convergence.