Page 283 - Applied Probability

12. Models of Recombination
271
In the count-location model, inequality (12.15) is equivalent to
(12.16)
<Q(1 − x 1 )Q(1 − x 2 ),
Q(1 − x 1 − x 2 )
2
2
λ 1 and x 2 =
d
λ 2 are the standardized map lengths of the
where x 1 =
d
intervals I 1 and I 2 , and Q(s) is the generating function of the total number
of chiasmata on the chromatid bundle. It is of some interest to characterize
the class C of discrete distributions for the chiasma count N guaranteeing
positive interference or at least noninterference. In general, we can say that
C
(a) is closed under convergence in distribution,
(b) is closed under convolution,
(c) contains all distributions whose generating functions Q(s) are log-
2
d
concave in the sense that ds 2 ln Q(s) ≤ 0,
(d) contains all distributions concentrated on the set {0, 1} or on the set
{1, 2, 3, 4}.
Properties (a) and (b) are trivial to deduce. Problems 8 and 9 address
properties (c) and (d). Property (d) is particularly relevant because most
chromatid bundles carry between one and four chiasmata. From these four
properties, we can build up a list of specific members of C. For instance,
C contains all binomial distributions and all distributions concentrated on
two adjacent integers. Compound Poisson distributions such as the negative
binomial exhibit negative interference rather than positive interference.
For the stationary renewal model, we allow equality in inequality (12.15)
and reexpress it for all y, z ≥ 0as
¯
¯
¯
F ∞ (y + z) ≤ F ∞ (y)F ∞ (z), (12.17)
where the right-tail probability
¯
F ∞ (x) = 1 − F ∞ (x)
1 ∞
= [1 − F(w + x)]dw. (12.18)
µ
0
¯
Log-concavity of F ∞ (x) is a sufficient condition for the submultiplicative
property (12.17). Indeed, the inequality
y+z d
¯
¯
ln F ∞ (y + z)= ln F ∞ (x)dx
0 dx
y z
d d
¯
¯
≤ ln F ∞ (x)dx + ln F ∞ (x)dx
0 dx 0 dx
¯
¯
=ln F ∞ (y)+ln F ∞ (z)