An Approximation to the Stationary Distribution of a Nearly Completely Decomposable Markov Chain and Its Error Bound

In Haviv (Ph.D. dissertation, Yale Univ., New Haven, CT, 1983) an approximation procedure for computing the stationary distribution of a nearly completely decomposable (NCD) Markov chain is suggested. There and in Haviv (this Journal, 7 (1986), pp. 589-593) the incurred error is analyzed. In particular, a series expansion for the error is developed. Courtois and Semal (J. Assoc. Comput. Mach., 31 (1984), pp. 804-825) independently of us, replaced this point approximation with a set of points. Using algebraic methods, they proved that the exact distribution lies in the convex set spanned by this set. We give a probabilistic interpretation for this set and then obtain their results in a more elementary way. We compute the convex combination leading to the exact distribution and develop a bound on it. Finally, we show how approximation to this convex combination leads to an error reduction in a cui`rent approximation. It is the first time that a probabilistic approach is made in order to analyze NCD Markov chains.