6.4 The Indeterminate Form at a= 0
All of the results established in this section assume that the mutation probability
parameter is strictly positive (a > 0), and thus are not applicable at a = 0. The reason is
apparent when Eq. 4.7 and the two-operator result in 4.17 (or the three-operator counter-
parts of Eq. 4.17 given by Eq. 4.23 and 4.27) are applied to Pj-II. It follows that the
row of the a 0' limit of I P-II corresponding to the one-operator absorbing state
nA E SA', nA : m is zero. That is, the only nonzero entry in row nA of the a -) 0 limit of
P- is the principal diagonal element
lim P2(nA I nA) = lim P(A nA)= Pl(nA I nA)= 1,
a0-- a-0+
which is cancelled by the corresponding principal diagonal element in -I. Thus,
Vm E S' : lim IP-II = 0,
a- 0+
and consequently Propositions 6.7 and 6.8 yield indeterminate forms. However, as dem-
onstrated in the following section, it is possible to verify that a limiting stationary distri-
bution vector exists for the time-homogeneous two and three-operator algorithms.