Suppose now that X is a Markov chain with finite state space and T is a finite Markov killing time for X. By obvious reductions there is no loss of generality involved in the following
Assumption. The state space of X is
where E is finite, 
is an absorbing state,
and
for all 
.
Let 
denote the substochastic semi-group of X restricted
to E, and view and all other operators as matrices indexed by E,
for example 
.
Then 
is just
is the usual Q-matrix of the chain killed at time T.
Recall that 
where 
is the operator of multiplication
by v.
It is clear that there exists b>0 such that
for all v with 
the matrix 
is invertible.
So
and the F-K formula (7) can be restated as follows:
There exists b>0 such that for all v with
For let 
.
Since 
,
formula (49) determines the joint moment generating function of
the 
.
The second expression in (49) for this m.g.f.
appears as formula (4) in Kingman [39], and
again in Puri [54].
Kingman noted as a consequence
that the joint m.g.f. is a ratio of two multilinear forms in v(x), , and that the marginal distribution of each 
is a mixture
of a point mass at zero and an exponential distribution on 
.
Kingman raised the problem, which is apparently still open,
of characterizing which joint distributions
can appear as the joint distributions of such
occupation times of a transient finite state chain.
For some study of particular examples see [36, 40].
Every killing time of a finite state chain is easily seen to be
of the form assumed in Proposition 1
for the killing rate function
where Q is the Q-matrix of the killed chain.
From (31)
so the assumption that 
for all x
implies 
.
Formula (30) now yields expressions for the 
conditional moments of 
given 
.
This leads to the following sharper form of the F-K formula for chains.
Formula (52) is a variant of Theorem 2.1 of Dynkin [18].
See also Sections I.27 of [57] and IV.22 of [56] for
related presentations.
Proof.
Formula (51) results from summing (30) weighted by -1/n!,
and using QG = -I.
To derive (52) from (51),
take 
to be a point mass at x and f to be the indicator
of a single point y, and cancel the common factor of k(y).
Formula (52) for 
can also be understood probabilistically
by consideration of 
; that is,
the probability that the chain starting at x ends with left limit
y at time T having survived the additional killing at rate v.
By conditioning on 
,
On the other hand, by conditioning on the same probability
equals
Comparing the two results yields (52). Note the parallel between (52) and the more obvious formula
where 
is the transition
function of the chain killed at time T, and 
is the same for
instead of T, namely the semigroup with Q-matrix
instead of Q, where
.
Compare with formula 2.6.6 of Itô-McKean [30] in Kac's original
Brownian setting.
A common generalization of these formulae
in an abstract setting could be given using h-processes,
but this is left to the reader.
