7Big theorems

II Probability and Measure

7.2 Central limit theorem

Theorem.

Let (

X

n

) be a sequence of iid random variables with

E

[

X

i

] = 0 and

E[X

2

1

] = 1. Then if we set

S

n

= X

1

+ ··· + X

n

,

then for all x ∈ R, we have

P

S

n

√

n

≤ x

→

Z

x

−∞

e

−y

2

/2

√

2π

dy = P[N(0, 1) ≤ x]

as n → ∞.

Proof.

Let

φ

(

u

) =

E

[

e

iuX

1

]. Since

E

[

X

2

1

] = 1

< ∞

, we can differentiate under

the expectation twice to obtain

φ(u) = E[e

iuX

1

], φ

0

(u) = E[iX

1

e

iuX

1

], φ

00

(u) = E[−X

2

1

e

iuX

1

].

Evaluating at 0, we have

φ(0) = 1, φ

0

(0) = 0, φ

00

(0) = −1.

So if we Taylor expand φ at 0, we have

φ(u) = 1 −

u

2

2

+ o(u

2

).

We consider the characteristic function of S

n

/

√

n

φ

n

(u) = E[e

iuS

n

/

√

n

]

=

n

Y

i=1

E[e

iuX

j

/

√

n

]

= φ(u/

√

n)

n

=

1 −

u

2

2n

+ o

u

2

n

n

.

We now take the logarithm to obtain

log φ

n

(u) = n log

1 −

u

2

2n

+ o

u

2

n

= −

u

2

2

+ o(1)

→ −

u

2

2

So we know that

φ

n

(u) → e

−u

2

/2

,

which is the characteristic function of a N (0, 1) random variable.

So we have convergence in characteristic function, hence weak convergence,

hence convergence in distribution.