7.5 The Stationary Distribution Limit
The zero mutation probability limits of Eq. 7.9 and Eq. 7.11 exist if the determinant
sums in the denominators are nonzero. In fact they are nonzero, as demonstrated in the
following. This argument is very similar in form to the development in Section 6.3 con-
cerning positivity of the stationary distribution. The essential step is demonstration of the
existence of a primitive stochastic matrix Q' which satisfies both
0 lim P, Q' and Q' lim P '.
a- 0 A C-40+ A
a-.o=+ O=0
If the two-operator algorithm is under consideration, then the elements of the
a -- 0' limit of PR are obtained by substituting the one-operator results in Eq. 4.2-5 into
Eq. 7.7. If the three-operator case is under consideration, then Eq. 4.22 and Eq. 4.24,25
are employed. In the following, the two-operator notation is employed.
Let Q' be generated from the a -- 0+ limit of P A' by replacing row m^ with the row
whose elements are given by
VmE S'-SA+ {mA :Q'( I mA)= > 0. Eq.7.12
N'-N+1
Thus, the row sum of row mA in Q' is 1. Since all remaining rows of Q' are identical to
those of the a -- 0+ limit of P ', and consequently have row sum 1 by Eq. 7.7, Q' is a
stochastic matrix. Additionally, it satisfies both
0 lim P-A' < Q' and Q' lim PA'.
a -0+ aO 0
Q' can be regarded as the state transition matrix of a fictitious Markov chain
defined on the state space S' S' + {m^}. Since Q(m, I m,) > 0, the fictitious Markov
chain is both periodic (Definition A9, Theorem A2) and primitive (Definition B1, Theo-
rem B1) provided that it is irreducible (Definitions A7 and A8, Theorem Al). Thus,
primitivity is established by demonstrating that every state m e S' S,' + {m^} is
accessible in some finite number of transitions from every state n S'- S' + {m,}.
Since all states in S'-SA'+ {mA} are accessible in one transition from m, (Eq. 7.12), it is