and consequently from Theorem A6, the chain is weakly ergodic if the sequence of con-
trol parameter values {a(k)} satisfies
( 2o(k) ML
k=l 1 + a(k)) )
Comparing this result to the known divergent series 'k-1, it follows that the Markov
chain is weakly ergodic if the sequence {(x(k)} satisfies
2C2a(k) ,
1 + a(k)
from which
1-k
_x(k) Ik.
1 + a(k)) 2
Using Eq. 4.13 to translate this result into an equivalent expression in pm(k) establishes
Proposition 8.1.
8.3 Strong Ergodicity
The mutation probability schedule bound advanced in Proposition 8.1 is also suffi-
cient to achieve strong ergodicity if it satisfies the condition on the sequence of vector
differences in Theorem A7. The required sequence of vectors can be selected as the
sequence of stationary distributions of the time-homogeneous Markov chains associated
with the parameter sequence {pm(k)} (or equivalently with the corresponding sequence
{a(k)}).
Section 4 establishes that a stationary distribution exists for the time-homogeneous
two and three-operator algorithms corresponding to every value of a satisfying a > 0.
Thus, associated with the sequence of control parameter values {a(k)} is a sequence of
vectors {qk} where q = q, evaluated at a = a(k). Further, based upon results established
in Section 6, Section 7 demonstrates that an a -> 0+ limiting stationary distribution exists
(Propositions 7.5 and 7.6), that the stationary distribution vector varies continuously for
all a satisfying a 2 0 (Proposition 7.7) and that its first derivative exists and is continuous