The collection of count-location models that are also stationary renewal models is very small [3]. Haldane’s homogeneous Poisson model is one example. Another is the trivial model with no chiasmata. The admixture of two such independent processes furnishes a third example subsuming the first two. To prove that this exhausts the possibilities, consider a count-location process that uniformly distributes its points on [0, 1]. Let qn denote the nth count probability. If the process is also a stationary renewal process, let F∞(x) be the distribution of X1 and F(x) be the distribution of the subsequent Xi, in the notation of Section 12.4. Show that F∞(x)=1 − ∞

n=0 qn(1 − x)

n and that F(x) = Pr(X2 ≤ x | X1 = y)

= 1 −

∞

n=0 nqn(1 − x − y)n−1

∞

n=0 nqn(1 − y)n−1 .

Use these identities and the identity F

∞(x) = [1−F(x)]/µ to demonstrate that φ(x) = ∞

n=0 nqn(1 − x)n−1 satisfies the functional equation φ(x)φ(y) = µ−1φ(x + y), where µ is the mean of F(x). Setting

ψ(x) = µφ(x), it follows that ψ(x)ψ(y) = ψ(x + y) and ψ(0) = 1.

The only decreasing function fitting this description is ψ(x) = e−βx for some β > 0.

Argue that this solution entails pn = βn−1e−β

µn! , n ≥ 1 1 − p0 = 1 − e−β

βµ .