5.3. Central Limit Theorem 343

Sincem′′(t) is continuous att=0andsinceξ→0asn→∞,wehave

lim

n→∞

[m′′(ξ)−σ^2 ]=0.

The limit proposition (5.2.16) cited in Section 5.2 shows that

lim

n→∞

M(t;n)=et

(^2) / 2
,
for all real values oft. This proves that the random variableYn=
√
n(Xn−μ)/σ
has a limiting standard normal distribution.
As cited in Remark 5.2.1, we say thatYnhas a limiting standard normal distri-
bution. We interpret this theorem as saying that whennis a large, fixed positive
integer, the random variableXhas an approximate normal distribution with mean
μand varianceσ^2 /n; and in applications we often use the approximate normal pdf
as though it were the exact pdf ofX. Also, we can equivalently state the conclusion
of the Central Limit Theorem as
√
n(X−μ)
D
→N(0,σ^2 ). (5.3.2)
This is often a convenient formulation to use.
One of the key applications of the Central Limit Theorem is for statistical infer-
ence. In Examples 5.3.1–5.3.6, we present results for several such applications. As
we point out, we made use of these results in Chapter 4, but we will also use them
in the remainder of the book.
Example 5.3.1 (Large Sample Inference forμ). LetX 1 ,X 2 ,...,Xnbe a ran-
dom sample from a distribution with meanμand varianceσ^2 ,whereμandσ^2
are unknown. LetXandSbe the sample mean and sample standard deviation,
respectively. Then
X−μ
S/
√
n
D
→N(0,1). (5.3.3)
To see this, write the left side as
X−μ
S/
√
n

(σ
S
)(X−μ)
σ/
√
n
.
Example 5.1.1 shows thatSconverges in probability toσand, hence, by the theo-
rems of Section 5.2, thatσ/Sconverges in probability to 1. Thus the result (5.3.3)
follows from the CLT and Slutsky’s Theorem, Theorem 5.2.5.
In Examples 4.2.2 and 4.5.3 of Chapter 4, we presented large sample confidence
intervals and tests forμbased on (5.3.3).
Some illustrative examples, here and below, help show the importance of this
version of the CLT.