Page 274 - Applied Probability

12. Models of Recombination
262
models. A renewal process is generated by the partial sums of a sequence
X 1 ,X 2 ,... of nonnegative, i.i.d. random variables. The first random point
on [0, ∞) occurs at X 1 , the second at X 1 + X 2, the third at X 1 + X 2 + X 3 ,
and so forth [8, 15, 25]. If the X i have common continuous distribution
function F(x) with mean µ, then for large x the interval (x, x+∆x) contains
approximately ∆x random points. The expected number of random points
µ
U(x) on the interval (0,x] coincides with the distribution function E(N [0,x] )
of the intensity measure. In the renewal theory literature, F(x) is said to
be the interarrival distribution, and U(x) is said to be the renewal
function. If the X i follow an exponential distribution, then the renewal
process collapses to a Poisson process with intensity λ = 1 and renewal
µ
x
function U(x)= .
µ
The identity U(x)= x characterizes a Poisson process. In general, we
µ
can achieve a uniform distribution for the intensity measure by passing
to a delayed renewal process where X 1 follows a different distribution
function F ∞ (x) than the subsequent X i . The appropriate choice of F ∞ (x)

turns out to have density F (x)= [1 − F(x)]/µ. Problem 3 sketches a
∞
proof of this fact. The delayed renewal process with density [1 − F(x)]/µ
is a stationary point process in the sense that it exhibits the same
stochastic behavior regardless of whether we begin observing it at 0 or
at some subsequent nonrandom point y> 0. In particular, the waiting
time until the next random point after y also follows the equilibrium
distribution F ∞ (x).
For more than two generations, geneticists have proposed various map
functions. The recent work of Zhao and Speed [32] clarifies which of these
map functions legitimately arise from point process models. They show
that any valid map function can be realized by constructing a stationary
renewal process for the underlying chiasma process. Implicit in this finding
is the fact that a map function does not uniquely define its chiasma process.
In exploring the map function problem, let us consider for the sake of
simplicity a map function θ = M(d) that is twice differentiable and whose
derivative satisfies the fundamental theorem of calculus in the form

d 2

M (d 2 ) − M (d 1 )= M (x)dx.

d 1
By virtue of Mather’s formula (12.1), it is clear that (a) M(0) = 0, (b)

M (d) ≥ 0, and (c) M (0) = 1. If at least one chiasma is certain on a

1
segment of infinite length, then it is also clear that (d) lim d→∞ M(d)= .
2
The final property (e) M (d) ≤ 0 is true but more subtle.

Property (e) can be proved by noting that formula (12.6) implies on one
hand that
y {1,2,3} − y {1,3}
1
= − 2Pr(N I 2 = 0)+2 Pr(N I 1 =0,N I 2 =0)
8