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.
Highlights
Let Q q i. j,i, j S be the stable, conservativeQ-matrix of a continuous-time Markov process on a countable state space S 0,1, 2, The Q-matrix satisfiesQ i, j 0, if j i,0 q i : Q i,i,if j i, andQ i, j 0 for all i S j SIn addition, we assume that Q is regular, which means there exists no non-trivial, non-negative solution x x j, j S to for some k 0 .Under these assumptions, the state transition probabilities of the process are given by the unique Q-func-tion F F t i, j,i, j S,t 0 which satisfies theKol- mogorov backward equations, d dt Ft QF t,t
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
We have investigated procedures based on the augmentation of state-space truncations for approximating the stationary distributions of positive recurrent, continuous-time Markov processes on countably infinite state spaces
Summary
Q-matrix of a continuous-time Markov process on a countable state space S 0,1, 2, The Q-matrix satisfies. We assume that Q is regular, which means there exists no non-trivial, non-negative solution x x j , j S to Under these assumptions, the state transition probabilities of the process are given by the unique Q-func-. By adding the discarded transition rates to n Q , we may produce a conservative Q-matrix n Q which generates a unique, honest, finite, continuous-time Markov process. Parallelling results for discrete-time chains in [7], we shall show that Markov processes constructed from finite perturbations of stochastically monotone processes are always dominated by some other stochastically monotone process This extends the class of processes for which our results are applicable.
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