Infinite Block-Structured Transition Matrices
and Their Properties
In this paper, we study Markov chains with infinite state
block-structured transition matrices,
whose states are partitioned into levels according to the
block structure, and various associated measures.
Roughly speaking, these measures involve first passage times or
expected numbers of visits to certain levels without hitting
other levels; they
are very important and often play a key role
in the study of a Markov chain.
Necessary and/or sufficient conditions are obtained for a Markov chain
to be ergodic, recurrent, or transient in terms of these measures.
Results are obtained for general irreducible Markov chains as well as
those with transition matrices possessing some block structure.
We also discuss the decomposition or the factorization of the characteristic
equations of these measures. In the scalar case, we locate the zeros of these
characteristic functions and therefore use these zeros to characterize
a Markov chain. Examples and various remarks are given to
illustrate some of the results.
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