In the analysis of Markov chains and processes, it is sometimes convenient to replace an unbounded state space with a "truncated" bounded state space. When such a replacement is made, one often wants to know whether the equilibrium behavior of the truncated chain or process is close to that of the untruncated system. For example, such questions arise naturally when considering numerical methods for computing stationary distributions on unbounded state space. In this paper, we use the principle of "regeneration" to show that the stationary distributions of "fixed state" truncations converge in great generality (in total variation norm) to the stationary distribution of the untruncated limit, when the untruncated chain is positive Harris recurrent. Even in countable state space, our theory extends known results by showing that the augmentation can correspond to an $r$-regular measure. In addition, we extend our theory to cover an important subclass of Harris recurrent Markov processes that include non-explosive Markov jump processes on countable state space.
翻译:在分析Markov 链条和过程时,有时以“ 疏松” 的封闭状态空间取代未封闭状态空间比较方便。 在进行这种替换时,人们往往想知道断线链条或过程的平衡行为是否接近未疏松系统。 例如,在考虑在无限制状态空间上计算固定分布的数字方法时,这类问题自然产生。 在本文中,我们使用“ 重新生成” 的原则来显示“ 固定状态” 串流的固定分布非常笼统( 完全变异规范) 与未疏松散限制的固定分布相融合, 而未疏松链条是正常态的哈里斯。 即使在可计数状态空间, 我们的理论扩展了已知结果, 显示增速可以与$- 经常计量相匹配。 此外, 我们扩展了我们的理论, 以涵盖哈里斯 斯 经常性马科夫 进程的重要子类, 其中包括可计算的状态空间上的非爆炸性马科夫 跳动过程 。