Slides Chapter 1. Measure Theory and Probability

1.9. THE CHARACTERISTIC FUNCTION
Proposition 1.21 (Properties of the c.f.)
Let ϕ(t) be the characteristic function associated with the d.f. F.
Then
(a) ϕ(0) = 1 (ϕ is non-vanishing at t = 0);
(b) |ϕ(t)| ≤ 1 (ϕ is bounded);
(c) ϕ(−t) = ϕ(t), ∀t ∈ IR, where ϕ(t) denotes the conjugate
complex of ϕ(t);
(d) ϕ(t) is uniformly continuous in IR, that is,
lim
h↓0
|ϕ(t+h)−ϕ(t)| = 0, ∀t ∈ IR.
Theorem 1.19 (c.f. of a linear transformation)
Let X be a r.v. with c.f. ϕ X (t). Then, the c.f. of Y = aX +b,
where a,b ∈ IR, is ϕ Y (t) = e itb ϕ X (at).
Example 1.22 (c.f. for some r.v.s)
Here we give the c.f. of some well known random variables:
(i) For the Binomial distribution, Bin(n,p), the c.f. is given by
ϕ(t) = (q +pe it ) n .
(ii) For the Poisson distribution, Pois(λ), the c.f. is given by
ϕ(t) = exp { λ(e it −1) } .
(iii) For the Normal distribution, N(µ,σ 2 ), the c.f. is given by
ϕ(t) = exp
{iµt− σ2 t 2 }
.
2
Lemma 1.2 ∀x ∈ IR, |e ix −1| ≤ |x|.
ISABEL MOLINA 47