Slides Chapter 1. Measure Theory and Probability

1.9. THE CHARACTERISTIC FUNCTION
We have seen that for absolutely continuous F, the c.f. ϕ(t) can
be expressed in terms of the p.d.f. f. However, it is possible to
express the p.d.f. f in terms of the c.f. ϕ(t)? The next theorem is
the answer.
Theorem 1.23 (Fourier transform of the c.f.)
If F is absolutely continuous and ϕ(t) is Riemann integrable in IR,
that is, ∫ ∞
−∞
|ϕ(t)|dt < ∞, then ϕ(t) is the c.f. corresponding to
an absolutely continuous r.v. with p.d.f. given by
f(x) = F ′ (x) = 1 ∫
e −itx ϕ(t)dt,
2π
where the last term is called the Fourier transform of ϕ.
In the following, we are going to study the c.f. of random variables
that share the probability symmetrically in IR + and IR − .
Definition 1.52 (Symmetric r.v.)
A r.v. X is symmetric if and only if X d = −X, that is, iff F X (x) =
F −X (x), ∀x ∈ IR.
Remark 1.15 (Symmetric r.v.)
Since ϕ is determined by F, X is symmetric iff ϕ X (t) = ϕ −X (t),
∀t ∈ IR.
Corolary 1.5 (Another characterization of a symmetric
r.v.)
X is symmetric iff F X (−x) = 1 − F X (x − ), where F X (x − ) =
P(X < x).
Corolary 1.6 (Another characterization of a symmetric
r.v.)
X is symmetric iff ϕ X (t) = ϕ X (−t), ∀t ∈ IR.
IR
ISABEL MOLINA 50