Page 278 - Applied Probability

12. Models of Recombination
266
space {0, 1, 2,...,r − 1} is summarized by the infinitesimal generator
r − 2
1
0
0

···
λs r
−λ
λ
0
0
1
···
 −λ(1 − s 1 ) λs 2 ··· λs r−1 r − 1 

Γ= .  . . . .  .
. . . . .
.  . . . . 
r − 1 0 0 ··· λ −λ
The equilibrium distribution for the chain has entries π m = 1 n>m n .
s
ω
Indeed, the balance condition πΓ= 0 reduces to
1 1
− s n λ(1 − s 1 )+ s n λ =0
ω ω
n>0 n>1
for row m = 0 and to
1 1 1
s n λs m+1 − s n λ + s n λ =0
ω ω ω
n>0 n>m n>m+1

for row m> 0. These equations follow from the identity s n =1.
n>0
The second Markov chain is identical to the first except that it has an
absorbing state 0 abs . In state 0 the chain moves to state 0 abs with transition
rate λs 1 . In state 1 it moves to state 0 abs instead of state 0 with transition
rate λ. If at most r − 1 o points can be skipped, then this second chain has
infinitesimal generator
0 1 ··· r − 2 r − 10 abs
0  −λλs 2 ··· λs r−1 λs r λs 1 
1  0 −λ ··· 0 0 λ 
.  . . . . . 
∆= . .  . . . . . . . .  .
 .
. 
r − 1  0 0 ··· λ −λ 0 
0 0 0 0 0
0 abs ···
As emphasized in Chapter 10, the entry p ij (t) of the matrix exponential
e tΓ provides the probability that the Poisson-skip process moves from state
i of the first Markov chain at time 0 to state j of the same chain at time t.
The entry q ij (t) of the matrix exponential e t∆ provides the probability that
the Poisson-skip process moves from state i of the first chain to state j of
the first chain without encountering a χ point during the interim. Because
∆ has the partition structure

Φ v
∆=
0 0
for Φ an r×r matrix and v a1×r column vector, one can easily demonstrate
that e t∆ has the corresponding partition structure with e tΦ as its upper