Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

334 Consistency and Limiting Distributions

As the following example shows, the converse of this theorem is not true.

Example 5.2.5. Take{Xn}to be the following sequence of degenerate random
variables. Forn=2meven,X 2 m=2+(1/(2m)) with probability 1. Forn=2m− 1
odd,X 2 m− 1 =1+(1/(2m)) with probability 1. Then the sequence{X 2 ,X 4 ,X 6 ,...}
converges in distribution to the degenerate random variableY = 2, while the se-
quence{X 1 ,X 3 ,X 5 ,...}converges in distribution to the degenerate random variable
W= 1. Since the distributions ofY andW are not the same, the sequence{Xn}
does not converge in distribution. Because all of the mass of the sequence{Xn}is
in the interval [1, 5 /2], however, the sequence{Xn}is bounded in probability.

One way of thinking of a sequence that is bounded in probability (or one that is
converging to a random variable in distribution) is that the probability mass of|Xn|
is not escaping to∞. At times we can use boundedness in probability instead of
convergence in distribution. A property we will need later is given in the following
theorem:

Theorem 5.2.7.Let{Xn}be a sequence of random variables bounded in probability
and let{Yn}be a sequence of random variables that converges to 0 in probability.
Then
XnYn
P
→ 0.
Proof:Let>0begiven. ChooseB >0 and an integerN such that

n≥N ⇒P[|Xn|≤B ]≥ 1 −.

Then

lim

n→∞

P[|XnYn|≥ ] ≤ lim

n→∞

P[|XnYn|≥,|Xn|≤B ]

+lim

n→∞

P[|XnYn|≥,|Xn|>B ]

≤ lim

n→∞

P[|Yn|≥/B ]+ =, (5.2.8)

from which the desired result follows.

5.2.2 Δ-Method.............................

Recall a common problem discussed in the last three chapters is the situation where

we know the distribution of a random variable, but we want to determine the

distribution of a function of it. This is also true in asymptotic theory, and Theorems

5.2.4 and 5.2.5 are illustrations of this. Another such result is called theΔ-method.

To establish this result, we need a convenient form of the mean value theorem

with remainder, sometimes called Young’s Theorem; see Hardy (1992) or Lehmann

(1999). Supposeg(x) is differentiable atx.Thenwecanwrite

g(y)=g(x)+g′(x)(y−x)+o(|y−x|), (5.2.9)

where the notationomeans

a=o(b) if and only ifab→0, asb→ 0.