5Fourier transform
II Probability and Measure
5.5 Properties of characteristic functions
We are now going to state a bunch of theorems about characteristic functions.
Since the proofs are not examinable (but the statements are!), we are only going
to provide a rough proof sketch.
Theorem.
The characteristic function
φ
X
of a distribution
µ
X
of a random
variable
X
determines
µ
X
. In other words, if
X
and
˜
X
are random variables
and φ
X
= φ
˜
X
, then µ
X
= µ
˜
X
Proof sketch.
Use the Fourier inversion to show that
φ
X
determines
µ
X
(
g
) =
E[g(X)] for any bounded, continuous g.
Theorem.
If
φ
X
is integrable, then
µ
X
has a bounded, continuous density
function
f
X
(x) = (2π)
−d
Z
φ
X
(u)e
−i(u,x)
du.
Proof sketch.
Let
Z ∼ N
(0
,
1) be independent of
X
. Then
X
+
√
tZ
has a
bounded continuous density function which, by Fourier inversion, is
f
t
(x) = (2π)
−d
Z
φ
X
(u)e
−|u|
2
t/2
e
−i(u,x)
du.
Sending
t →
0 and using the dominated convergence theorem with dominating
function |φ
X
|.
The next theorem relates to the notion of weak convergence.
Definition
(Weak convergence of measures)
.
Let
µ,
(
µ
n
) be Borel probability
measures. We say that
µ
n
→ µ
weakly if and only if
µ
n
(
g
)
→ µ
(
g
) for all
bounded continuous g.
Similarly, we can define weak convergence of random variables.
Definition
(Weak convergence of random variables)
.
Let
X,
(
X
n
) be random
variables. We say
X
n
→ X
weakly iff
µ
X
n
→ µ
X
weakly, iff
E
[
g
(
X
n
)]
→ E
[
g
(
X
)]
for all bounded continuous g.
This is related to the notion of convergence in distribution, which we defined
long time ago without talking about it much. It is an exercise on the example
sheet that weak convergence of random variables in
R
is equivalent to convergence
in distribution.
It turns out that weak convergence is very useful theoretically. One reason is
that they are related to convergence of characteristic functions.
Theorem.
Let
X,
(
X
n
) be random variables with values in
R
d
. If
φ
X
n
(
u
)
→
φ
X
(u) for each u ∈ R
d
, then µ
X
n
→ µ
X
weakly.
The main application of this that will appear later is that this is the fact
that allows us to prove the central limit theorem.
Proof sketch.
By the example sheet, it suffices to show that
E
[
g
(
X
n
)]
→ E
[
g
(
X
)]
for all compactly supported
g ∈ C
∞
. We then use Fourier inversion and
convergence of characteristic functions to check that
E[g(X
n
+
√
tZ)] → E[g(X +
√
tZ)]
for all
t >
0 for
Z ∼ N
(0
,
1) independent of
X,
(
X
n
). Then we check that
E[g(X
n
+
√
tZ)] is close to E[g(X
n
)] for t > 0 small, and similarly for X.