338 Consistency and Limiting DistributionsIf this sum is substituted foret√
2 /nin the last expression forMY
n(t), it is seen thatMYn(t)=(
1 −
t^2
n+
ψ(n)
n)−n/ 2
,where
ψ(n)=√
2 t^3 eξ(n)
3√
n
−√
2 t^3
√
n
−2 t^4 eξ(n)
3 n
.Sinceξ(n)→0asn→∞, then limψ(n) = 0 for every fixed value oft. In accordance
with the limit proposition cited earlier in this section, we have
lim
n→∞
MYn(t)=et(^2) / 2
for all real values oft. That is, the random variableYn=(Zn−n)/
√
2 nhas a
limiting standard normal distribution.
Figure 5.2.2 displays a verification of the asymptotic distribution of the stan-
dardizedZn.Foreachvalueofn=5, 10 ,20 and 50, 1000 observations from a
χ^2 (n)-distribution were generated, using the R commandrchisq(1000,n).Each
observationznwas standardized asyn=(zn−n)/
√
2 nand a histogram of these
yns was computed. On this histogram, the pdf of a standard normal distribution is
superimposed. Note that atn= 5, the histogram ofynvalues is skewed, but asn
increases, the shape of the histogram nears the shape of the pdf, verifying the above
theory. These plots are computed by the R functioncdistplt. In this function, it
is easy to change values ofnfor further such plots.
Example 5.2.8(Example 5.2.7, Continued).In the notation of the last example,
we showed that
√
n
[
1
√
2 n
Zn−
1
√
2
]
D
→N(0,1). (5.2.17)
For this situation, though, there are times when we are interested in the square
root ofZn.Letg(t)=
√
tand letWn=g(Zn/(
√
2 n)) = (Zn/(
√
2 n))^1 /^2 .Notethat
g(1/
√
2) = 1/ 21 /^4 andg′(1/
√
2) = 2−^3 /^4. Therefore, by the Δ-method, Theorem
5.2.9, and (5.2.17), we have
√
n
[
Wn− 1 / 21 /^4
]
→DN(0, 2 −^3 /^2 ). (5.2.18)
EXERCISES
5.2.1.LetXndenote the mean of a random sample of sizenfrom a distribution
that isN(μ, σ^2 ). Find the limiting distribution ofXn.
5.2.2.LetY 1 denote the minimum of a random sample of sizenfrom a distribution
that has pdff(x)=e−(x−θ),θ<x<∞, zero elsewhere. LetZn=n(Y 1 −θ).
Investigate the limiting distribution ofZn.