Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
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<strong>1.</strong>9. THE CHARACTERISTIC FUNCTION<br />
(IR,B). The characteristic function (c.f.) of X is<br />
ϕ(t) = E [ e itX] ∫<br />
= e itX dP, t ∈ IR.<br />
Remark <strong>1.</strong>11 (The c.f. is determined by the d.f.)<br />
ThefunctionY t = g t (X) = e itX isacompositionoftwomeasurable<br />
functions, X(ω) <strong>and</strong> g t (x), that is, Y t = g t ◦ X. Then, Y t is<br />
measurable <strong>and</strong> by the Theorem of change of integration space,<br />
the c.f. is calculated as<br />
∫ ∫<br />
ϕ(t) = e itX dP = (g t ◦X)dP<br />
∫Ω<br />
∫ Ω ∫<br />
= g t dP X = g t (x)dF(x) = e itx dF(x),<br />
Ω<br />
IR<br />
where P X is the probability induced by X <strong>and</strong> F is the d.f. associated<br />
with P X . Observe that the only thing that we need to obtain<br />
ϕ is the d.f., F, that is, ϕ is uniquely determined by F.<br />
Remark <strong>1.</strong>12 Observe that:<br />
• ∫ IR eitx dF(x) = ∫ IR cos(tx)dF(x)+i∫ IR sin(tx)dF(x).<br />
• Since |cos(tx)| ≤ 1 <strong>and</strong> |sin(tx)| ≤ 1, then it holds:<br />
∫ ∫<br />
|cos(tx)| ≤ 1, |sin(tx)| ≤ 1<br />
IR<br />
Ω<br />
<strong>and</strong> therefore, |cos(tx)| <strong>and</strong> |sin(tx)| are integrable. This<br />
means that ϕ(t) exists ∀t ∈ IR.<br />
• Many properties of the integral of real functions can be translatedtotheintegralofthecomplexfunctione<br />
itx . Inpractically<br />
allcases, theresultisastraightforwardconsequenceofthefact<br />
that to integrate a complex values function is equivalent to integrate<br />
separately the real <strong>and</strong> imaginary parts.<br />
IR<br />
IR<br />
ISABEL MOLINA 46