Page 276 - Applied Probability

12. Models of Recombination
264
Conversely, if we postulate the existence of a map function M(d) satisfy-
ing properties (a) through (e), then equation (12.10) defines a valid density
function f(x). Indeed, property (e) indicates that f(x) ≥ 0, and properties
1
(a) and (d) indicate that

2

(e), this last fact entails lim x→∞ M (x) = 0. Thus, the calculation
∞
" 0 ∞ M (x)dx = . In view of properties (b) and

f(x)dx = M (0) − lim M (x)
0 x→∞
=1
verifies that f(x) has total mass 1. Using f(x) to construct a stationary
renewal process yields a map function matching M(d) and proves Zhao and
Speed’s converse.
Example 12.4.1 Kosambi’s Map Function
Kosambi’s map function [16] M(d)= 1 tanh(2d) has first two derivatives
2
4
M (d) =

(e 2d + e −2d 2
)
e 2d − e −2d

M (d) = −16 .
(e 2d + e −2d 3
)
From these expressions it is clear that properties (a) through (e) are true.
Taking µ = 1 in equation (12.10) yields
2
e 2x − e −2x
f(x) = 16 .
(e 2x + e −2x 3
)
12.5 Poisson-Skip Model
The Poisson-skip process is a particularly simple stationary renewal
model that is generated by a Poisson process with intensity λ and a skip
distribution s n on the positive integers. Random Poisson points are divided
into o points and χ points; o points are “skipped” to reach χ points, which
naturally correspond to chiasmata. At each χ point, one independently
chooses with probability s n to skip n − 1 o points before encountering
the next χ point. This recipe creates a renewal process with interarrival
distribution
∞ ∞ m
(λx) −λx
F(x) = s n e .
m!
n=1 m=n
The Poisson tail probability ∞ (λx) m e −λx appearing in this formula is
m=n m!
the probability that the nth random point to the right of the current χ point