Page 287 - Applied Probability

12. Models of Recombination
275
If G(x) is chosen to make the delayed renewal process stationary, then
show that
x
x

U(x − y)dG(y),
= G(x)+
µ
0
" ∞ −λx (12.21)
where µ is the mean of F(x). If dH(λ)= e dH(x) denotes
%
0
the Laplace transform of the distribution function H(x) defined on
(0, ∞), also verify the identity
1 − dF(λ)
%
dG(λ) = .
%
µÐ
Finally, prove that the Laplace transform of the density 1 [1 − F(x)]
µ
matches dG(λ).
%
4. Show that Felsenstein’s [9] map function
1 e 2(2−γ)d − 1
θ = (12.22)
2 e 2(2−γ)d − γ +1
arises from a stationary renewal model when 0 ≤ γ ≤ 2. Kosambi’s
map function is the special case γ = 0. Why does (12.22) fail to give
a legal map function when γ> 2? Note that at γ = 2 we define
θ = d by l’H opital’s rule.
2d+1
5. Continuing Problem 4, prove that Felsenstein’s map function has in-
verse

1 1 − 2θ
d = ln .
2(γ − 2) 1 − 2(γ − 1)θ
6. The Carter and Falconer [4] map function has inverse
1 −1 −1
−1
M (θ) = [tan (2θ) + tanh (2θ)].
4
Prove that the map function satisfies the differential equation
4

M (d) = 1 − 16M (d)
with initial condition M(0) = 0. Deduce from these facts that M(d)
arises from a stationary renewal model.
2πi
7. Fix a positive integer m, and let w m = e m be the principal mth root
of unity. For each integer j, define the segmental function m α j (x)
of x to be the finite Fourier transform
m−1
1 xw k −jk
m α j (x) = e m w m .
m
k=0
These functions generalize the hyperbolic trig functions cosh(x) and
sinh(x). Prove the following assertions: