Page 93 - Elements of Distribution Theory

P1: JZP
052184472Xc03 CUNY148/Severini May 24, 2005 2:34

3.2 Basic Properties 79

(2)
(2)
Now suppose that ϕ (0) exists. Writing ϕ (0) in terms of the first derivative ϕ and

then writing ϕ in terms of ϕ,wehave

ϕ(h) − 2ϕ(0) + ϕ(−h)
(2)
ϕ (0) = lim .
h→0 h 2
Now,
ϕ(h) − 2ϕ(0) + ϕ(−h) = E[exp(ihX) − 2 + exp(−ihX)]
2
2
= E{[exp(ihX/2) − exp(−ihX/2)] }= E{[2i sin(hX/2)] }.
Hence,

2
[i sin(hX/2)] 2
(2)
ϕ (0) = lim E X
h→0 (hX/2) 2
and
sin(hX/2) 2
2
(2)
|ϕ (0)|= lim E X .
h→0 (hX/2) 2
By Fatou’s Lemma,
sin(hX/2) 2 sin(hX/2) 2
2 2
lim E X ≥ E lim inf X
h→0 (hX/2) 2 h→0 (hX/2) 2
and since
sin(t)
lim = 1
t→0 t
we have that
2 (2)
E(X ) ≤|ϕ (0)|
2
so that E(X ) < ∞.
The general case is similar, but a bit more complicated. Suppose that ϕ (2m) (0) exists for
some m = 1, 2,.... Since

ϕ (2m−2) (h) − 2ϕ (2m−2) (0) + ϕ (2m−2) (−h)
(2m)
ϕ (0) = lim ,
h→0 h 2
it may be shown by induction that
2m
1 j 2m
(2m)
ϕ (0) = lim (−1) ϕ(( j − m)h).
h→0 h 2m j
j=0
Furthermore,
2m
2m
j 2m j 2m
(−1) ϕ(( j − m)h) = E (−1) exp[i( j − m)hX]
j j
j=0 j=0
2m
= E{[exp(ihX/2) − exp(−ihX/2)] }
2m
= E{[2i sin(hX/2)] }.