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.