Abstract

this paper, we are mainly concerned with the characterization of the fundamental matrix Z(") of the perturbed chain. In the case of a singular perturbation, Z( ) also has a discontinuity at = 0. Moreover, in this case, jjZ(")jj !1 as " ! 0 and Z(") admits a Laurent series expansion [132, 134, 66, 67] Z(") = 1 " s Z s + ::: + 1 " Z 1 + Z 0 + "Z 1 + :::; " 6= 0: (3.47) Note that s, the order of the pole, is nite and s N . We denote the singular and the regular parts of the fundamental matrix expansion (3.47) by Z S (") and Z R ("), respectively. Schweitzer has also obtained formulae for the matrices Z k . However, they are rather complicated and their computation requires to handle large size matrices (cf. [132]). In sequel, we propose a di erent approach that allows us to compute more eciently the coecients of the Laurent series (3.47). This approach can be considered as a particular realisation of the general scheme proposed in Section 2.4 for the perturbation analysis of group inverses. Our method leads to the operations with matrices of small dimensions. For example, immediately after the rst stage of the reduction process one handles aggregated Markov chains with no more than m states (m being the number of ergodic classes). This would, typically, constitute a drastic reduction of the dimension. In addition, for the case of a linear perturbation we provide a simple formula for the regular part Z R ( ), which readily simpli es to the usual formula [131] if the perturbation is regular. We introduce the deviation matrices (or reduced resolvents) H and H(") for the original and perturbed chains: H def = Z P and H(") def = Z(") P ("): We immediately have H S (") = Z S (") and H R (") = Z R (") P ("): (3.48) The fundamental matri...

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