8.2 A Weak Ergodicity Bound
The following paragraphs propose and then verify a mutation probability parameter
bound sufficient to ensure that the Markov chain of the corresponding nonstationary sim-
ple genetic algorithm is weakly ergodic (Definition All). The bound applies to both the
two and three-operator algorithms, and it appears in Proposition 8.1 below.
Proposition 8.1: The mutation probability bound given by
1 --
pm(k)2 k M
is sufficient to ensure weak ergodicity of the corresponding nonstationary (two or three-
operator) simple genetic algorithm Markov chain.
This result is established by using the lower bounds on the two and three-operator
conditional probabilities in Eq. 4.21 and 4.31 with Definitions Al l and A12 and Theo-
rems A5 and A6. Applying the lower bound in Eq. 4.21 and 4.31 to T,(.) of Definition
A 12 and Theorem A5 yields
T,(P) = 1 min min(P(m I n,), P(m | n))
Ii D2 m
++a i m
1+aa jl +a
+2a i
1+a)
Thus,
(P)) 2a
>-+(aP)