336 Consistency and Limiting Distributionswhereopis interpreted as in (5.2.10). Rearranging, we have
√
n(g(Xn)−g(θ)) =g′(θ)√
n(Xn−θ)+op(√
n|Xn−θ|).Because (5.2.14) holds, Theorem 5.2.6 implies that
√
n|Xn−θ|is bounded in prob-
ability. Therefore, by Theorem 5.2.8,op(
√
n|Xn−θ|)→0, in probability. Hence,
by (5.2.14) and Theorem 5.2.1, the result follows.Illustrations of the Δ-method can be found in Example 5.2.8 and the exercises.5.2.3 MomentGeneratingFunctionTechnique............
To find the limiting distribution function of a random variableXnby using the
definition obviously requires that we knowFXn(x) for each positive integern.But
it is often difficult to obtainFXn(x) in closed form. Fortunately, if it exists, the
mgf that corresponds to the cdfFXn(x) often provides a convenient method of
determining the limiting cdf.
The following theorem, which is essentially Curtiss’ (1942) modification of a
theorem of L ́evy and Cram ́er, explains how the mgf may be used in problems of
limiting distributions. A proof of the theorem is beyond of the scope of this book. It
can readily be found in more advanced books; see, for instance, page 171 of Breiman
(1968) for a proof based on characteristic functions.
Theorem 5.2.10.Let{Xn}be a sequence of random variables with mgfMXn(t)
that exists for−h<t<hfor alln.LetXbe a random variable with mgfM(t),
which exists for|t|≤h 1 ≤h.Iflimn→∞MXn(t)=M(t)for|t|≤h 1 ,thenXn→DX.
In this and the subsequent sections are several illustrations of the use of Theorem
5.2.10. In some of these examples it is convenient to use a certain limit that is
established in some courses in advanced calculus. We refer to a limit of the form
lim
n→∞[
1+
b
n+
ψ(n)
n]cn
,wherebandcdo not depend uponnand where limn→∞ψ(n)=0. Then
lim
n→∞[
1+b
n+ψ(n)
n]cn
= lim
n→∞(
1+b
n)cn
=ebc. (5.2.16)For example,
lim
n→∞(
1 −
t^2
n+
t^2
n^3 /^2)−n/ 2
= lim
n→∞(
1 −
t^2
n+
t^2 /√
n
n)−n/ 2
.Hereb=−t^2 ,c=−^12 ,andψ(n)=t^2 /√
n. Accordingly, for every fixed value oft,the limit iset
(^2) / 2
.