Page 277 - Applied Probability

12. Models of Recombination
265
∞

lies within a distance x of the current χ point. If we let ω =
ns n be
n=1
the mean number of points until the next χ point, then Wald’s formula [15]
shows that F(x) has mean
λ
is
m
∞
∞
λ
λ ω . The density of the equilibrium distribution
(λx)

−λx

[1 − F(x)] = s n 1 − e
ω ω m!
n=1 m=n
∞ n−1 m
λ (λx) −λx
= s n e .
ω m!
n=1 m=0
According to equation (12.1), the map function for the Poisson-skip
model boils down to
1 λ ∞ !
θ = 1 − [1 − F(y)]dy
2 ω x
1 λ ∞ ∞ n−1 (λy) m −λy !

= 1 − s n e dy .
2 ω x m!
n=1 m=0
Because successive integrations by parts yield
m
m k
(λy) −λy 1 (λx) −λx
∞
e dy = e ,
m! λ k!
x
k=0
it follows that
∞ n−1 m k
1 1 (λx) −λx !
θ = 1 − s n e
2 ω k!
n=1 m=0 k=0
∞ n−1
1 e −λx (λx) k !
= 1 − s n (n − k) .
2 ω k!
n=1 k=0
To calculate gamete probabilities under the Poisson-skip model, it is
helpful to consider two associated Markov chains. The state space for the
first chain is {0, 1, 2,...}. When the chain is in state 0, the most recent
point encountered was a χ point. When it is in state i> 0, it is must
pass exactly i − 1 o points before encountering the next χ point. Thus,
if the chain is currently in state 0, then it moves to state n − 1,n > 1,
with transition rate λs n . This transition mechanism implies that the chain
decides how many o points to skip simultaneously with moving to the next
point. When the chain decides to skip no o points, it remains in state 0.
If the chain is currently in state n> 0, then it falls back to state n − 1
with transition rate λ. These are the only moves possible. If at most r − 1
o points can be skipped, then the motion of the chain on the reduced state