where
D= P,(0 1 21)3 [P(0 1 12)3 + 3P1(0 I 12)2P (1 12)] +
P,(0| 12)3 [Pl(1 21)3 + 3P(0 I 21)P1(1 1 21)2] +
P,(1 I 12)3[Pi(1 1 21)3 +3P(1 I 21)2Pl(0 1 21) +
P,(1 21)3[P,(0 1 12)3+3P1(1 1 12)P,(0 12)2].
The Eq. 9.33 counterpart is
[ 2R(0)]3 [R(0)3 + 6R(0)2R(1)] + R(0)3 [R( )3 + 6R(0)R(1)2]
qo(30) =
and Eq. 9.35
S[2R(1)]3 [R(1)3 + 6R(1)2R(0)] + R(1)3 [R(0)3 + 6R(1)R(0)2]
qo(03) =
where
D' = [2R(0)]3 [R(0)3 + 6R(0)2R(1)] + R(0)3 [R(1)3 + 6R(0)R(1)2]
+[2R(1)]3 [R(1)3 + 6R(1)2R(0)] + R(1)3 [R(0)3 + 6R(1)R(0)2].
Again, the three-operator case yields an identical result.
These examples suggest two very significant conjectural features of the limiting sta-
tionary distribution behavior. First, only order 2 monomial terms survive in the determi-
nant expansions of the M=2 case and only order 6 terms survive for M=3. These facts
lead to the supposition that in general, only order Mx(N'-N) terms survive. In the M=2
case, Mx(N'-N) = 2x(3-2) = 2 while for M=3, Mx(N'-N) = 3x(4-2) = 6. If this supposi-
tion is correct, then the polynomial forms required for evaluating the stationary distribu-
tion zero mutation probability limit by Propositions 7.5 and 7.6 are homogeneous order
Mx(N'-N) polynomials in the P(i I n)'s. Presumably, the corresponding property (i.e.
homogeneous order Mx(N'-1) order polynomial forms) applies to the general case repre-
sented by Propositions 6.7 and 6.8.