Stochastic Block-Monotonicity in the Approximation of the Stationary Distribution of Infinite Markov Chains

Markov chains, whose transition matrices reveal a certain type of block-structure, find many applications in various areas. Examples include Markov chains of GI/M/1 type and M/G/1 type, and more generally Markov chains of Toeplitz type. Some Markov chains without a block-repeating structure can be also included; for example, level-dependent-quasi-birth-and-death (LDQBD) processes. In analyzing this type of Markov chains, one may find that properties and/or probabilistic measures described or expressed by probability transition blocks from level to level often play a dominating role, while detailed transitions between states within the same level (block) are less important. In this paper, we introduce the concept of block-monitonicity and apply this notion to dealing with Markov chains possessing a block structure. A successful application in approximating stationary probability vectors of an infinite-state Markov chain is provided. We also hope that more applications of this concept can be exposed in the future.
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