-MLa + (a2)
(+ )7- n = nA
MC
VnA E SA' T(n I A)= ()L nE S(nA)' Eq. 7.5
ne S'- S(nA'-
(1 + )ML
where the exponent s of ca in the order expression for the general term is an integer satis-
fying s 2 2. The elements in columns nA and n E S(nA)' are first order in a while the ele-
ments in all other columns are at least second order.
Eq. 7.5 applies to every absorbing state row of I Pj 11 as well if m e SA'. If
Pi I is being considered where mA e SA', then row mA contains -1 at its principal
diagonal and zeros elsewhere. In that case Eq. 7.5 only applies to the absorbing state rows
n, e SA' {mA}. Exactly N- 1 such rows exist in Pi I .
By applying Eq. 7.5 and these observations to Proposition 7.1, it follows that the
lowest order term with nonzero coefficient which can conceivably exist in the numerator
polynomial of Pm^-i1 is the order cN-' term. Similar reasoning reveals that the corre-
sponding lowest order term with nonzero coefficient for I P,- II with m e SA' is the order
aN term. If the coefficient of the order a"N- term in the numerator polynomial of IP^ -1
is indeed nonzero, and if the corresponding coefficients for all such m, have the same
algebraic sign, then the required limiting value of q can be expressed in terms of these
nonzero coefficients via substitution into Proposition 6.7. These conditions are in fact
satisfied as demonstrated below.
7.4 Reformulation of Propositions 6.7 and 6.8
The next step in this development is the definition of some auxiliary matrices
related to P and Pin and the reformulation of Propositions 6.7 and 6.8 in terms of them.
The new matrices, designated P(mA)' and PW respectively, are derived by coalescing
each of the N 1 absorbing state columns nA e S,' {nmA of P- 11 and P^ -1I with its
neighboring nonabsorbing state columns, n e S(nA)'. Specifically, let ^ be derived