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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