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B. Kirstein, “Monotonicity and Comparability of Time-Homogeneous Markov Processes with Discrete State Space,” Statistics, Vol. 7, No. 1, 1976, pp. 151-168.

has been cited by the following article:

• TITLE: Convergence of Invariant Measures of Truncation Approximations to Markov Processes

  AUTHORS: Andrew G. Hart, Richard L. Tweedie

  KEYWORDS: Invariant Measure; Truncation Approximation; Augmentation; Exponential Ergodicity; Stochastic Monotonicity; Markov Process

  JOURNAL NAME: Applied Mathematics, Vol.3 No.12A, December 31, 2012

  ABSTRACT: Let Q be the Q-matrix of an irreducible, positive recurrent Markov process on a countable state space. We show that, under a number of conditions, the stationary distributions of the n × n north-west corner augmentations of Q converge in total variation to the stationary distribution of the process. Two conditions guaranteeing such convergence include exponential ergodicity and stochastic monotonicity of the process. The same also holds for processes dominated by a stochastically monotone Markov process. In addition, we shall show that finite perturbations of stochastically monotone processes may be viewed as being dominated by a stochastically monotone process, thus extending the scope of these results to a larger class of processes. Consequently, the augmentation method provides an attractive, intuitive method for approximating the stationary distributions of a large class of Markov processes on countably infinite state spaces from a finite amount of known information.