Page 288 - Applied Probability

12. Models of Recombination
276
(a) m α j (x)= m α j+m (x).
m−1
k=0 m α k (x) m α j−k (y).

(b) m α j (x + y)=
j+km
x
∞

(c) m α j (x)=
k=0 (j+km)!

d
(d) dx m α j (x) = m α j−1 (x). for 0 ≤ j ≤ m − 1.
m
d
(e) Consider the differential equation dx m f(x)= kf(x) with initial
conditions d j j f(0) = c j for 0 ≤ j ≤ m − 1, where k and the c j
dx
are constants. Show that
m−1
j 1
f(x) = c j k − m α j (k m x).
m
j=0
(f) The differential equation d m m f(x)= kf(x)+ g(x) with initial
dx
j
d
conditions dx j f(0) = c j for 0 ≤ j ≤ m − 1 has solution
x
m−1 1
f(x) = k − m m α m−1 [k m (x − y)]g(y)dy
0
m−1
j 1

+ c j k − m α j (k m x).
m
j=0
1
(g) lim x→∞ e −x m α j (x)= m .
(h) In a Poisson process of intensity 1, e −x m α j (x) is the probability
that the number of random points on [0,x] equals j modulo m.
(i) Relative to this Poisson process, let N x count every mth random
point on [0,x]. Then N x has probability generating function
m−1
j 1
−x
P(s) = e s − m α j (s m x).
m
j=0
(j) Furthermore, N x has mean
m−1
x e −x
E(N x) = − j m α j (x).
m m
j=0

x m−1
(k) lim x→∞ E(N x ) − = − .
m 2m
8. In the count-location model, suppose that the count distribution has
a log-concave generating function Q(s). Prove that the model exhibits
positive or no interference [23].