Proposition 6.8: The components of the stationary distribution can be expressed in the
alternative form
I P-II -IPR-l
q,(m) -
O (IP-II-I|P-II)
ne S'
where as before P, and PN are derived from P by replacing the rows indexed by m and n
~~T
respectively with the row vector 0 .
6.3 Positivity of the Stationary Distribution Components
Strict positivity of the stationary distribution components can be deduced from
Theorem B4 and the form of Pn. Every element of PR in every row other than row m is
identical to the corresponding element of P, while those in row m are zero. This is
expressed in the nonnegative matrix notation of Appendix B as 0 < P- 0, the value of the determinant I P-II satisfies
Vm e S' : (1) IPR-I| Oand
(2) the algebraic sign of P_-ll is (-1)N.
An immediate consequence of Proposition 6.9 is that both numerator and denomi-
nator of the expression for q.(m) in Proposition 6.7 are nonzero and have identical alge-
braic sign. Strict positivity of the stationary distribution components follows from these
observations. That is, Vm e S': q,,(m) > 0.