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.