Page 88 - Elements of Distribution Theory

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

74 Characteristic Functions

and that
∞

lim g(x) cos(tx) dx = 0.
t→∞
−∞
These follow immediately from the Riemann-Lebesgue Lemma.
Now consider part (ii). Note that it suffices to show that

1 T ∞
g(x 0 ) = lim g(x) cos(t(x − x 0 )) dx dt (3.2)
2π T →∞ −T −∞
and
T ∞
lim g(x) sin(t(x − x 0 )) dx dt = 0. (3.3)
T →∞ −T −∞
Consider (3.3). Changing the order of integration,

T ∞ ∞ T
g(x) sin(t(x − x 0 )) dx dt = g(x) sin(t(x − x 0 )) dt dx.
−T −∞ −∞ −T
Equation (3.3) now follows from the fact that, for T > 0,

T
sin(t(x − x 0 )) dt = 0.
−T
Now consider (3.2). Again, changing the order of integration, and using the change-of-
variable u = x − x 0 ,

T ∞ ∞ T
g(x) cos(t(x − x 0 )) dx dt = g(x) cos(t(x − x 0 )) dt dx
−T −∞ −∞ −T

∞ sin(Tu)
= g(u + x 0 ) du.
u
−∞
It now follows from Section A3.4.11 that
∞ sin(Tu)

lim g(u + x 0 ) du = g(x 0 ).
T →∞ u
−∞
The following theorem applies this result to characteristic functions and shows that the
distributionfunctionofareal-valuedrandomvariablemaybeobtainedfromitscharacteristic
function, at least at continuity points of the distribution function.

Theorem 3.3. Let X denote a real-valued random variable with distribution function F
and characteristic function ϕ.IfF is continuous at x 0 , x 1 ,x 0 < x 1 , then
1 T exp{−itx 0 }− exp{−itx 1 }
F(x 1 ) − F(x 0 ) = lim ϕ(t) dt.
2π T →∞ −T it

Proof. Fix x. Define

h(y) = F(x + y) − F(y).