Proposition 7.3, which concerns the first derivative ofq,, can also be extended to
include the limiting case. The extension requires easily obtainable counterparts of Eq.
7.1-3 developed for IP' -I' and Eq. 7.9. The Eq. 7.1 counterpart is
(1+a) tM( '-) PF = 0-(a)', Eq. 7.13
and that for Eq. 7.2 is
6 -.(a)' + O(a)
6( O-- m = mA e SA'
(m)= () S Eq. 7.14
O(a)
( -me S -SA '
6(a)' + 0(a) A
where E(a)' is the polynomial counterpart (summed over n, e SA') of E(a) in Eq. 7.2.
Differentiating Eq. 7.14 with respect to a yields a rational function with denominator
polynomial [%(a)' + O(a)]2, whose a -- 0 limit is nonzero by Proposition 7.4, Eq. 7.13
and the definition of 6(a)'. Proposition 7.8 below follows from Proposition 7.3 and these
observations.
Proposition 7.8: The components of the first derivative ofq,, with respect to a possess
limits as a -- 0'.
Thus, a zero mutation probability limit exists for the time-homogeneous two and
three-operator algorithm variants. The limit is represented by Propositions 7.5 and 7.6.
Further, Propositions 7.7 and 7.8 establish some useful ancillary results concerning the
stationary distribution behavior at the point a = 0. These latter results are employed in the
following section in establishing strong ergodicity of the inhomogeneous genetic algo-
rithm Markov chain. Propositions 7.5 and 7.6 are used in Section 9 to develop a method-
ology for representing the stationary distribution limit.