Proposition 7.6: The components of lim q, exist and can be expressed alternatively as
a -0*
lim P(mA)'- -I
a 0
__ [ ---o-''---=-- -----' m m= mA e SA
lim q,(m) = qo(m) = lim I P(nA n)' -' -
aE 0 n0A E SA' a 0-0
0 } me S'- SA'
An immediate consequence of Propositions 7.4 and 7.5 is strict positivity of the
zero mutation probability limiting stationary distribution components for all absorbing
state rows. That is, VmA e S : q0(mA) > 0. The argument is analogous to that at the
conclusion of Section 6.3 concerning strict positivity of all stationary distribution compo-
nents when a > 0. This result is anticipated by the simulation results in Section 5.5. A
consequence is that the required limiting behavior for direct application of the simulated
annealing convergence theory to the genetic algorithm model does not follow. However,
the results displayed in Section 5.5 and developments produced in Section 9.3 suggest
that the limiting distribution can be made arbitrarily close to the desired limiting behav-
ior.
Since the a -- 0+ limit of the stationary distribution exists, the definition of q, can
be extended to include the point a = 0. That is
qjao= 90= lim qa
a -0+
where the values of the required limits are provided by Proposition 7.5. Proposition 7.7
below follows from this extended definition of q, and Proposition 7.2.
Proposition 7.7: For all a 2 0, the components of q, are continuous rational functions of
the independent variable a.