M!
P3(m P I n) =1 n )mP(i )" Eq. 4.25
Fm(i)! ies
iE S
= 1xn P3(i In)m(i)
(mB iE S
and
P= [P3(mI n)]. Eq. 4.26
These results are developed in a fashion analogous to Eq. 4.14-4.16. From them, it fol-
lows that the three-operator Markov chain is time-homogeneous if both the mutation and
crossover probabilities are fixed. In general it is not time-homogeneous.
From Eq. 4.22, 4.24 and 4.25, it follows that
lim P3(i n) = P2'(i | n) Eq. 4.27
a .)+
and lim P3(m | n) = P'(m n).
a -+ 0
Also, from Eq. 4.23-4.25, the three-operator analogs of Eq. 4.18-19 apply
II(i, iA)
ca
P3(i I n= + Eq. 4.28
P3(m I nA) = L -ML Eq. 4.29
Additionally, since
aI L P2'( I n) < Y P2(j I n) x a" (ij)- P'(j I n),
jeS jES jES
the three-operator analogs of Eq. 4.20-21 follow from Eq. 4.24-25, i.e.
Vi E S, VE S' : P,(i n) < Eq. 4.30
1 +ac I+a
and
Vm,ne S': a P(m P In)( J(EI Eq. 4.31
Im I+a II I +a