(Eq. 2.6) are established. First, note that if the stationary distribution of a Markov chain
in the sequence exists, then a function g(C(i),T) corresponding to that Markov chain
exists such that
g(C(i), T)
Vi c E: qr(i) g(C(T) Eq. 2.8
Sg(C(j), T)
J
where g satisfies
(1) Vie E, VT>0 : g(C(i),T) > 0
Sg(C(i), T)Gj(T)A,(T) = Eq. 2.9
(2) Vj EE ij
g(C(i), T) Y_ Gj(T)Ai(T)
i*J
This can be deduced by noting that the uniquely determining conditions on q expressed in
Theorem A3 are met by g satisfying Eq. 2.8 and 2.9. Eq. 2.9 is called the global balance
equation. Close examination reveals that it is exactly the necessary condition for equilib-
rium state occupancy. A more restrictive condition, in which the balance holds for every
pair of states on a pair-wise basis is called the detailed balance equation.
It can be shown that the following additional constraints on g guarantee conver-
gence of the stationary distribution to the optimal (i.e. to Eq. 2.6) [MiRo85]. Note that
Eq. 2.10(2) requires an exponential form.
(1) lim g(A,T) = 0 A>0
T-0 [00 A<0
g(A,,T) Eq. 2.10
(2) ( = g(A A, T)
g(A2, T)
(3) VT > 0 : g(0,T)= 1
Collectively, Eq. 2.8-2.10 provide a set of sufficient conditions on Gj(T) and Aj(T)
to assure convergence of the stationary distribution to Eq. 2.6. The key condition, the
global balance equation, is implicit however, and thus is very difficult to apply. Neverthe-
less, it can be shown [LaAa87] that if G,(T) and Aj(T) defined by Eq. 2.4 and Eq. 2.5 are