59.18. THE CENTRAL LIMIT THEOREM 1923

Now let λ X (∂A) = 0 for A a Borel set.

λ X (int(A)) ≤ lim infn→∞

λ Xn (int(A))≤ lim infn→∞

λ Xn (A)≤

lim supn→∞

λ Xn (A) ≤ lim supn→∞

λ Xn

(A)≤ λ X

(A).

But λ X (int(A)) = λ X(A)

by assumption and so limn→∞ λ Xn (A) = λ X (A) as claimed. Thisproves the theorem.

As an application of this theorem the following is a version of the central limit theoremin the situation in which the limit distribution is multivariate normal. It concerns a sequenceof random vectors, {Xk}∞

k=1, which are identically distributed, have finite mean m, andsatisfy

E(|Xk|2

)< ∞. (59.18.41)

Definition 59.18.7 For X a random vector with values in Rp, let

FX (x)≡ P({

X j ≤ x j for each j = 1,2, ..., p})

.

Theorem 59.18.8 Let {Xk}∞

k=1 be random vectors satisfying 59.18.41, which are inde-pendent and identically distributed with mean m and positive definite covariance ≡E((X−m)(X−m)∗

). Let

Zn ≡n

∑j=1

X j−m√n

. (59.18.42)

Then for Z∼Np (0, ) ,limn→∞

FZn (x) = FZ (x) (59.18.43)

for all x.

Proof: The characteristic function of Zn is given by

φ Zn(t) = E

(eit·∑n

j=1X j−m√

n

)=

n

∏j=1

E

(e

it·(

X j−m√

n

)).

By Taylor’s theorem applied to real and imaginary parts of eix, it follows

eix = 1+ ix− f (x)x2

2

where | f (x)|< 2 andlimx→0

f (x) = 1.

Denoting X j as X, this implies

eit·(

X−m√n

)= 1+ it·X−m√

n− f

(t·(

X−m√n

))(t·(X−m))2

2n