Proof of Central Limit Theorem
H. Krieger, Mathematics 157, Harvey Mudd College
Spring, 2005
Preliminary Inequalities:
In order to utilize the result (sometimes called
the continuity theorem) that convergence in distribution is equivalent to point
wise convergence of the corresponding characteristic functions, we need the fol
lowing estimates about Taylor expansions of exponential functions.
1. If
u
≥
0, then
0
≤
e

u

1 +
u
≤
u
2
/
2
.
2. If
t
is real, then

e
it

1

it
 ≤ 
t

2
/
2
and

e
it

1

it

(
it
)
2
/
2
 ≤ 
t

3
/
6
.
Central Limit Theorem:
Let
{
X
n
}
be a sequence of i.i.d. (independent
identically distributed) random variables with common mean 0 and common
variance 1.
Then, if
Z
∼
N
(0
,
1) and
S
n
=
X
1
+
X
2
+
· · ·
+
X
n
, we have
S
n
/
√
n
→
Z
in distribution as
n
→ ∞
. In other words, for every
x
∈
R
,
lim
n
→∞
P
X
1
+
X
2
+
· · ·
+
X
n
√
n
≤
x
¶
=
1
√
2
π
Z
x
∞
e

u
2
/
2
du.
Proof:
Let
ˆ
F
be the characteristic function of the common distribution of the
{
X
n
}
. Then for every
t
∈
R
, the characteristic function of
S
n
/
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 CARRIKER
 Central Limit Theorem, Probability theory, Complex number, Sn, t2 /2n

Click to edit the document details