Page 100 - Elements of Distribution Theory

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

86 Characteristic Functions

In Theorem 3.8 it was shown that if |ϕ(t)|→ 0fast enough as |t|→∞ for
∞

|ϕ(t)| dt
−∞
to be finite, then the distribution is absolutely continuous. The following theorem gives a
partial converse to this: if X has an absolutely continuous distribution, then the characteristic
function of X approaches 0 at ±∞. Furthermore, the smoothness of p is related to the rate
at which the characteristic function approaches 0.

Theorem 3.9. Let X denote a real-valued random variable with characteristic function ϕ.
If the distribution of X is absolutely continuous with density function p then
ϕ(t) → 0 as |t|→∞. (3.7)

If p is k-times differentiable with
∞

(k)
|p (x)| dx < ∞,
−∞
then
−k
|ϕ(t)|= o(|t| ) as |t|→∞.

Proof. If X has an absolutely continuous distribution with density p, then the characteristic
function of X is given by

∞

ϕ X (t) = exp(itx)p(x) dx.
−∞
Hence, (3.7) follows directly from Theorem 3.2.
Suppose p is differentiable. Then, using integration-by-parts,
1 ∞ 1 ∞

ϕ X (t) = exp(itx)p(x) − exp(itx)p (x) dx.
it −∞ it −∞
Clearly, p(x) must approach 0 at ±∞ and, since exp(itx)is bounded,
1 ∞

ϕ X (t) =− exp(itx)p (x) dx.
it
−∞
If p satisfies

∞

|p (x)| dx < ∞,
−∞
then Theorem 3.2 applies to p so that

∞
exp(itx)p (x) dx → 0as |t|→∞.

−∞
Hence,
−1
|ϕ X (t)|= o(|t| )as |t|→∞.
The results for the higher-order derivatives follow in a similar manner.