342 Consistency and Limiting DistributionsTheorem 5.3.1(Central Limit Theorem).LetX 1 ,X 2 ,...,Xndenote the observa-
tions of a random sample from a distribution that has meanμand positive variance
σ^2. Then the random variableYn=(
∑n
i=1Xi−nμ)/√
nσ=√
n(Xn−μ)/σcon-
verges in distribution to a random variable that has a normal distribution with mean
zero and variance 1.
Proof: For this proof, additionally assume that the mgfM(t)=E(etX)existsfor
−h<t<h. If one replaces the mgf by the characteristic functionφ(t)=E(eitX),
which always exists, then our proof is essentially the same as the proof in a more
advanced course which uses characteristic functions.
The function
m(t)=E[et(X−μ)]=e−μtM(t)
also exists for−h<t<h.Sincem(t)isthemgfforX−μ, it must follow that
m(0) = 1,m′(0) =E(X−μ) = 0, andm′′(0) =E[(X−μ)^2 ]=σ^2 .ByTaylor’s
formula there exists a numberξbetween 0 andtsuch that
m(t)=m(0) +m′(0)t+m′′(ξ)t^2
2=1+m′′(ξ)t^2
2
.Ifσ^2 t^2 /2 is added and subtracted, thenm(t)=1+σ^2 t^2
2+[m′′(ξ)−σ^2 ]t^2
2(5.3.1)Next considerM(t;n), whereM(t;n)=E[
exp(
t∑
Xi−nμ
σ√
n)]= E[
exp(
t
X 1 −μ
σ√
n)
exp(
t
X 2 −μ
σ√
n)
···exp(
t
Xn−μ
σ√
n)]= E[
exp(
t
X 1 −μ
σ√
n)]
···E[
exp(
t
Xn−μ
σ√
n)]={
E[
exp(
tX−μ
σ√
n)]}n=[
m(
t
σ√
n)]n
, −h<t
σ√
n<h.In equation (5.3.1), replacetbyt/σ√
nto obtainm(
t
σ√
n)
=1+t^2
2 n
+[m′′(ξ)−σ^2 ]t^2
2 nσ^2
,where nowξis between 0 andt/σ
√
nwith−hσ√
n<t<hσ√
n. Accordingly,M(t;n)={
1+t^2
2 n
+[m′′(ξ)−σ^2 ]t^2
2 nσ^2}n
.