Slides Chapter 1. Measure Theory and Probability

1.9. THE CHARACTERISTIC FUNCTION
Remark 1.13 (Proposition 1.21 does not determine a
c.f.)
If ϕ(t) is a c.f., then Properties (a)-(d) in Proposition 1.21 hold
but the reciprocal is not true, see Example 1.23.
Theorem 1.20 (Moments are determined by the c.f.)
If the n-th moment of F, α n = ∫ IR xn dF(x), is finite, then
(a) Then-thderivativeofϕ(t)att = 0existsandsatisfiesϕ n) (0) =
i n α n .
(b) ϕ n) (t) = i n ∫ IR eitx x n dF(x).
Corolary 1.4 (Series expansion of the c.f.)
If α n = E(X n ) exists ∀n ∈ IN, then it holds that
ϕ X (t) =
∞∑
n=0
(it) n
α n , ∀t ∈ (−r,r),
n!
where (−r,r) is the radius of convergence of the series.
Example 1.23 (Proposition 1.21 does not determine a
c.f.)
Consider the function ϕ(t) = 1 , t ∈ IR. This function verifies
1+t 4
properties (a)-(d) in Proposition 1.21. However, observe that the
first derivative evaluated at zero is
∣ ϕ ′ (0) =
−4t 3 ∣∣∣t=0
∣ = 0.
(1+t 4 ) 2
The second derivative at zero is
∣ ϕ ′ (0) =
−12t 2 (1+t 4 ) 2 +4t 3 2(1+t 4 )4t 3 ∣∣∣t=0
∣
= 0.
(1+t 4 ) 4
ISABEL MOLINA 48