Page 97 - Elements of Distribution Theory

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

3.3 Further Properties of Characteristic Functions 83

m
In Theorem 3.5, it was shown that the existence of expected values of the form E(X ), for
m = 1, 2,..., is related to the smoothness of the characteristic function of X at 0. Note that
m
finiteness of E(|X| ) depends on the rates at which F(x) → 1as x →∞ and F(x) → 0
as x →−∞; for instance, if X has an absolutely continuous distribution with density p,
m
m
then E(|X| ) < ∞ requires that |x| p(x) → 0as x →±∞. That is, the behavior of the
distribution function of X at ±∞ is related to the smoothness of its characteristic function
at 0.
The following result shows that the behavior of |ϕ(t)| for large t is similarly related to
the smoothness of the distribution function of X.

Theorem 3.8. Consider a probability distribution on the real line with characteristic func-
tion ϕ.If
∞

|ϕ(t)| dt < ∞
−∞
then the distribution is absolutely continuous. Furthermore, the density function of the
distribution is given by

1 ∞
p(x) = exp(−itx)ϕ(t) dt, x ∈ R.
2π −∞
Proof. By Lemma A2.1, | exp(it) − 1|≤|t| so that for any x 0 , x 1 ,

| exp(−itx 0 ) − exp(−itx 1 )|=| exp(−itx 0 )||1 − exp{−it(x 1 − x 0 )}|≤|t||x 1 − x 0 |.
(3.5)
Hence, for any T > 0,
T
exp{−itx 0 }− exp{−itx 1 } ∞
lim ϕ(t) dt ≤|x 1 − x 0 | |ϕ(t)| dt. (3.6)

T →∞ −T it −∞
Let F denote the distribution function of the distribution; it follows from Theorem 3.3,
along with Equation (3.6), that, for any x 0 < x 1 at which F is continuous,
1 T exp{−itx 0 }− exp{−itx 1 }
F(x 1 ) − F(x 0 ) = lim ϕ(t) dt
2π T →∞ −T it
1 ∞ exp{−itx 0 }− exp{−itx 1 }
= ϕ(t) dt.
2π it
−∞
Since
| exp(−itx 0 ) − exp(−itx 1 )|≤|t||x 1 − x 0 |,
it follows that
1 ∞
|F(x 1 ) − F(x 0 )|≤ |ϕ(t)| dt |x 1 − x 0 |≤ M |x 1 − x 0 |
2π
−∞
for some constant M.
Now let x and y, y < x,be arbitrary points in R, i.e., not necessarily continuity points
of F. There exist continuity points of F, x 0 and x 1 , such that x 0 ≤ y < x ≤ x 1 and

|x 1 − x 0 |≤ 2|x − y|.