characteristic function

characteristic function

Let

X

be a

random variable

. The

characteristic function

of

X

is a

function

φ

X

:

ℝ

→

ℂ

defined by

φ

X

⁢

(

t

)

=

E

⁢

e

i

⁢

t

⁢

X

=

E

⁢

cos

⁡

(

t

⁢

X

)

+

i

⁢

E

⁢

sin

⁡

(

t

⁢

X

)

,

that is,

φ

X

⁢

(

t

)

is the

expectation

of the random variable

e

i

⁢

t

⁢

X

.

Given a random vector

X

¯

=

(

X

1

,

…

,

X

n

)

, the characteristic function of

X

¯

, also called

joint characteristic function

of

X

1

,

…

,

X

n

,
is a function

φ

X

¯

:

ℝ

n

→

ℂ

defined by

φ

X

¯

⁢

(

t

)

=

E

⁢

e

i

⁢

t

¯

⋅

X

¯

,

where

t

¯

=

(

t

1

,

…

,

t

n

)

and

t

¯

⋅

X

¯

=

t

1

⁢

X

1

+

⋯

+

t

n

⁢

X

n

(the dot product.)

Remark.

If

F

X

is the

distribution function

associated to

X

, by the
properties of expectation we have

φ

X

⁢

(

t

)

=

∫

ℝ

e

i

⁢

t

⁢

x

⁢

𝑑

F

X

⁢

(

x

)

,

which is known as the Fourier-Stieltjes transform of

F

X

, and provides an alternate
definition of the characteristic function. From this, it is clear that the characteristic
function depends only on the distribution function of

X

, hence one can define the characteristic
function associated to a

distribution

even when there is no random variable involved.
This implies that two random variables with the same distribution must have the same
characteristic function. It is also true that each characteristic function determines
a unique distribution; hence the , since it characterizes the distribution function (see property 6.)

Properties

1.

The characteristic function is

bounded

by

1

, i.e.

|

φ

X

⁢

(

t

)

|

≤

1

for all

t

;

2.

φ

X

⁢

(

0

)

=

1

;

3.

φ

X

⁢

(

t

)

¯

=

φ

X

⁢

(

-

t

)

, where

z

¯

denotes the complex conjugate of

z

;

4.

φ

X

is

uniformly continuous

in

ℝ

;

5.

If

X

and

Y

are

independent

random variables, then

φ

X

+

Y

=

φ

X

⁢

φ

Y

;

6.

The characteristic function determines the distribution function; hence,

φ

X

=

φ

Y

if and only if

F

X

=

F

Y

.
This is a

consequence

of the

inversion

: Given a random variable

X

with characteristic function

φ

and distribution function

F

, if

x

and

y

are

continuity

points of

F

such that

x

<

y

, then

F

⁢

(

x

)

-

F

⁢

(

y

)

=

1

2

⁢

π

⁢

lim

s

→

∞

⁡

∫

-

s

s

e

-

i

⁢

t

⁢

x

-

e

-

i

⁢

t

⁢

y

i

⁢

t

⁢

φ

⁢

(

t

)

⁢

𝑑

t

;

7.

A random variable

X

has a symmetrical distribution (i.e. one such that

F

X

=

F

-

X

)
if and only if

φ

X

⁢

(

t

)

∈

ℝ

for all

t

∈

ℝ

;

8.

For

real numbers

a

,

b

,

φ

a

⁢

X

+

b

⁢

(

t

)

=

e

i

⁢

t

⁢

b

⁢

φ

X

⁢

(

a

⁢

t

)

;

9.

If

E

⁢

|

X

|

n

<

∞

, then

φ

X

has

continuous

n

-th

derivatives

and

d

k

⁢

φ

X

d

⁢

t

k

⁢

(

t

)

=

φ

X

(

k

)

⁢

(

t

)

=

∫

ℝ

(

i

⁢

x

)

k

⁢

e

i

⁢

t

⁢

x

⁢

𝑑

F

X

⁢

(

x

)

,

1

≤

k

≤

n

.

Particularly,

φ

X

(

k

)

⁢

(

0

)

=

i

k

⁢

E

⁢

X

k

; characteristic functions are

similar

to

moment generating functions

in this sense.

Similar properties hold for joint characteristic functions.
Other important result related to characteristic functions is the Paul Lévy
continuity theorem.

Title

characteristic function

Canonical name

CharacteristicFunction1

Date of creation

2013-03-22 13:14:28

Last modified on

2013-03-22 13:14:28

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Definition

Classification

msc 60E10

Synonym

joint characteristic function

Related topic

MomentGeneratingFunction

Related topic

CumulantGeneratingFunction

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