Page 121 - Elements of Distribution Theory

P1: JZP
052184472Xc04 CUNY148/Severini May 24, 2005 2:39

4.3 Laplace Transforms and Moment-Generating Functions 107

Hence,

n n+1

|hx|
j

exp{itx} exp{ihx}− (ihx) /j! ≤
(n + 1)!
j=0
and

n j
n+1
∞
h |h| γ n+1
j
ϕ X (t + h) − (ix) exp{itx} dF X (x) ≤ ,
j! (n + 1)!

j=0 −∞
where ϕ X denotes the characteristic function of X.
Note that

∞ (k)
k
(ix) exp{itx} dF X (x) = ϕ (t).
X
−∞
Hence,

n j n+1

h |h|
( j) γ n+1
ϕ X (t + h) − ϕ (t) ≤ , n = 1, 2,....
X
j! (n + 1)!
j=0

It follows that
∞ j
h ( j)
ϕ X (t + h) = ϕ (t), |h| <δ. (4.2)
X
j!
j=0
Applying the same argument to Y shows that ϕ Y , the characteristic function of Y, satisfies
∞ j
h ( j)
ϕ Y (t + h) = ϕ (t), |h| <δ. (4.3)
Y
j!
j=0
Taking t = 0 and using the fact that
(k) k k (k)
ϕ (0) = E(X ) = E(Y ) = ϕ (0),
X Y
it follows that
ϕ X (t) = ϕ Y (t), |t| <δ
and also that
(k) (k)
ϕ (t) = ϕ (t), k = 1, 2,..., |t| <δ.
X Y
Taking t = δ/2in (4.2) and (4.3) shows that
∞ j
h ( j)
ϕ X (δ/2 + h) = ϕ (δ/2), |h| <δ
X
j!
j=0
and
∞ j
h ( j)
ϕ Y (δ/2 + h) = ϕ (δ/2), |h| <δ.
Y
j!
j=0
Since
(k) (k)
ϕ (δ/2) = ϕ (δ/2), k = 1, 2,...,
X Y