Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

352 Consistency and Limiting Distributions

Σwhich is positive definite. Assume that the common moment generating function

M(t)exists in an open neighborhood of 0 .Let

Yn=

1

√

n

∑n

i=1

(Xi−μ)=

√

n(X−μ).

ThenYnconverges in distribution to aNp( 0 ,Σ)distribution.

Proof:Lett∈Rpbe a vector in the stipulated neighborhood of 0. The moment
generating function ofYnis

Mn(t)=E

[

exp

{

t′

1

√

n

∑n

i=1

(Xi−μ)

}]

= E

[

exp

{

1

√

n

∑n

i=1

t′(Xi−μ)

}]

= E

[

exp

{

1

√

n

∑n

i=1

Wi

}]

, (5.4.10)

whereWi=t′(Xi−μ). Note thatW 1 ,...,Wnare iid with mean 0 and variance
Var(Wi)=t′Σt. Hence, by the simple Central Limit Theorem,

1

√

n

∑n

i=1

Wi

D

→N(0,t′Σt). (5.4.11)

Expression (5.4.10), though, is the mgf of (1/

√

n)

∑n
i=1Wievaluated at 1. There-
fore, by (5.4.11), we must have

Mn(t)=E

[

exp

{

(1)

1

√

n

∑n

i=1

Wi

}]

→e^1

(^2) t′Σt/ 2
=et
′Σt/ 2
.
Because the last quantity is the moment generating function of aNp( 0 ,Σ) distri-
bution, we have the desired result.
SupposeX 1 ,X 2 ,...,Xnis a random sample from a distribution with mean
vectorμand variance-covariance matrixΣ.LetXnbe the vector of sample means.
Then, from the Central Limit Theorem, we say that
Xnhas an approximateNp
(
μ,^1 nΣ
)
distribution. (5.4.12)
A result that we use frequently concerns linear transformations. Its proof is
obtained by using moment generating functions and is left as an exercise.
Theorem 5.4.5.Let{Xn}be a sequence ofp-dimensional random vectors. Suppose
Xn
D
→ N(μ,Σ).LetAbe anm×pmatrix of constants and letb be anm-
dimensional vector of constants. ThenAXn+b
D
→N(Aμ+b,AΣA′).