SECTION 7
THE ZERO MUTATION PROBABILITY STATIONARY DISTRIBUTION LIMIT
7.1 Overview
In Section 4.3.1, it is established that the time-homogeneous one-operator genetic
algorithm Markov chain possesses a stationary distribution but that it is not unique. In
Sections 4.3.2 and 4.3.3, it is established that the time-homogeneous two and three-
operator counterparts possess unique stationary distributions provided a > 0, and Section
6 formulates the existence and uniqueness argument into a rational function expression
for the unique solution. Since the two-operator state transition matrix approaches its one-
operator counterpart as a -- 0O (Eq. 4.17) and since the three-operator algorithm exhibits
the corresponding behavior with respect to the P2'(i n)s (Eq. 4.23), a question which
naturally arises from these observations is whether an a( 0+ limiting distribution exists
for the two and three-operator algorithms. (If such a limit exists, then it is necessarily
unique). This section answers that question affirmatively and also confirms the observa-
tion made in Section 5.5 that the limiting distribution is nonzero for all states correspond-
ing to uniform populations (absorbing states).
The approach taken here is to transform the expressions for q,,(m) in Propositions
6.7 and 6.8 into equivalent expressions which yield determinate forms at a = 0. The result
requires transforming P and P- into related matrices but with the states corresponding to
uniform populations (one-operator absorbing states) coalesced into adjacent nonuniform
population states. The development is tedious and involves some additional notation.
7.2 Functional Form of the Stationary Distribution
Before proceeding with the limiting case development which is the primary purpose
of this section, it is convenient to establish some intermediate results concerning the