MATH 589 Assignment 4 Due April 7, 2004
(1) Show that Yhas a symmetric stable distribution iff φY(t)=e−b|t|α, where b≥0 and
0<α≤2.
(2) Prove that the stable distributions are all absolutely continuous.
(3) For each n≥1, let Xn
1,Xn
2,... ,Xn
nbe i.i.d. r.v.’s, each having the uniform distribution
on [−n, n]. Define
Yn=M
n
m=1
sign Xn
m
|Xn
m|p,
where M>0 and p>1/2. If the Xn
mare positions of masses Mdistributed at random
on [−n, n], then Ynis the gravitational force exerted on a unit mass at the origin,
assuming an inverse pth power law.
(a) Show that the c.f. of Ynis
φn(t)=1−1
n∞
0
[1 −cos(Mt
xp)] dx −g(n)n
,
where g(n)→0asn→∞.
(b) Show that φn(t)→f(t) = exp(−∞
0[1 −cos(Mt
xp)] dx)asn→∞.
(c) Make the change of variable y=|Mt|1/px−1to show that f(t) is of the form e−d|t|α
where d>0 and 0 <α<2.
(d) Show that e−d|t|αwhere d>0 and 0 ≤α≤2 is a c.f.
(4) If λ≥0 and φ(t) is a c.f. of a random variable, show that eλ(φ(t)−1) is an infinitely
divisible c.f.
Hint: for problems 1 and 2, use the formula for the general form of the c.f. of a stable
law.
