lim P2(i ) = P1(i I n)
a -0+
and Eq. 4.17
lim P2(m n) = P,(m | n).
a- 0+
The rows of the state transition matrix corresponding to the one-operator absorbing
states have an especially simple form. Let iA e S be the solution represented in the
absorbing state n A SA'. Then, from Eq. 4.14,
Mx R(iA) x o 1I(iA)
P2(i I n ) = Eq. 4.18
(1 + a)L x MxR(iA)
H(i, iA)
(1 +a)L
Thus, from Eq. 4.15,
(M ) (Xrnm(i) x 1l(i, iA)
P2(m I nA)= M.) (+-x)ML Eq. 4.19
Since the reward function, R, is strictly positive by hypothesis, and since
Vi,j E S : 0 H(i,j) < L, it follows that for a in the range 0 < (a < 1, then
oa" L n(j) x Rj) Y n(j)xR(j) x a"(i'J) n(j) x R(j),
je S je S je S
and consequently from Eq. 4.14 that
Sa 1 L
Vie S, Vne S':- P2(i |n) ( Eq. 4.20
Using Eq. 4.20 in Eq. 4.15 yields
a 1 )ML
Vm, ne S' : P2(m | n) Eq. 4.21
m 1l+a m l+a
From the lower bound in Eq. 4.21, the final requirement of Theorem A3 (irreduc-
ibility) is fulfilled and the Markov chain for the time-homogeneous two-operator simple
genetic algorithm possesses a unique stationary distribution, q,, given by