On the Perturbation of Markov Chains with Nearly Transient States
On the Perturbation of Markov Chains with Nearly Transient States
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1998-10-15
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To Appear in Numerische Mathematik
Let $A$ be an irreducible stochastic matrix of the form [ A = \bmx{cc} A_{11} & E_{12} \ A_{21} & A_{22} \emx. ] If $E_{22}$ were zero, the states corresponding to $A_{22}$ would be transient in the sense that if the steady state vector $y\trp$ is partitioned conformally in the form $(y_1\trp ; y_2\trp)$ then $y_2\trp = 0$. If $E_{22}$ is small, then $y_2\trp$ will be small, and the states are said to be nearly transient. It this paper it is shown that small relative perturbations in $A_{11}$, $A_{21}$, and $A_{22}$, though potentially larger than $y_2\trp$, induce only small relative perturbations in $y_2\trp$. (Also cross-referenced as UMIACS-TR-92-14)