Slides Chapter 1. Measure Theory and Probability

1.9. THE CHARACTERISTIC FUNCTION
(IR,B). The characteristic function (c.f.) of X is
ϕ(t) = E [ e itX] ∫
= e itX dP, t ∈ IR.
Remark 1.11 (The c.f. is determined by the d.f.)
ThefunctionY t = g t (X) = e itX isacompositionoftwomeasurable
functions, X(ω) and g t (x), that is, Y t = g t ◦ X. Then, Y t is
measurable and by the Theorem of change of integration space,
the c.f. is calculated as
∫ ∫
ϕ(t) = e itX dP = (g t ◦X)dP
∫Ω
∫ Ω ∫
= g t dP X = g t (x)dF(x) = e itx dF(x),
Ω
IR
where P X is the probability induced by X and F is the d.f. associated
with P X . Observe that the only thing that we need to obtain
ϕ is the d.f., F, that is, ϕ is uniquely determined by F.
Remark 1.12 Observe that:
• ∫ IR eitx dF(x) = ∫ IR cos(tx)dF(x)+i∫ IR sin(tx)dF(x).
• Since |cos(tx)| ≤ 1 and |sin(tx)| ≤ 1, then it holds:
∫ ∫
|cos(tx)| ≤ 1, |sin(tx)| ≤ 1
IR
Ω
and therefore, |cos(tx)| and |sin(tx)| are integrable. This
means that ϕ(t) exists ∀t ∈ IR.
• Many properties of the integral of real functions can be translatedtotheintegralofthecomplexfunctione
itx . Inpractically
allcases, theresultisastraightforwardconsequenceofthefact
that to integrate a complex values function is equivalent to integrate
separately the real and imaginary parts.
IR
IR
ISABEL MOLINA 46