recipe employed in its construction, it follows that the square matrix pzi. thus defined has
mA
dimension N' N + 1 and that its elements are given by
Eq. 7.7
0 n =mA
n # mA
SP(m I n) m S' SA' +
Vi, n A
S' -SA'+mA} m n)'= {mA + S(mA) SA".
n # mA
P(m n) + P(nA n) me S(nA)'
nA e SA' {mA
Careful examination of Eq. 7.7 reveals that the transformation by which P A' is gen-
erated from PE preserves all row sums. Thus, P5,' is very similar in form to PP It is
derived from a (fictitious) row stochastic matrix by setting a specified row (row mA) to
zero.
If the preceding steps are repeated for I Pn-iI where m SA', except that all N
absorbing state columns n e SA' are coalesced rather than just the N 1 columns
nA E SA' {mA}, a result very similar in form to Eq. 7.6 obtains. That is,
(-MLa)N I Q,'I O(Ca)
P-- = Q = + Eq.7.8
(1 + a)MLN (1 + a)MLN'
where I Q'|I is the order N' N principal minor of I Q5j generated by deleting the N
absorbing state row/column pairs and s is an integer satisfying s 2 N + 1. The nonabsorb-
ing state row m contains -1 at its principal diagonal location and zeros elsewhere.
Substitution of Eq. 7.6 and 7.8 into Proposition 6.7 yields a form more amenable to
examination of the a -4 0' limiting stationary distribution. The two cases in SA' and
m S'- SA' must be distinguished. Then, after some straightforward algebra,