Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

5.2. Convergence in Distribution 333

Theorem 5.2.4.SupposeXnconverges toXin distribution andgis a continuous

function on the support ofX.Theng(Xn)converges tog(X)in distribution.

An often-used application of this theorem occurs when we have a sequence of
random variablesZnwhich converges in distribution to a standard normal random
variableZ. Because the distribution ofZ^2 isχ^2 (1), it follows by Theorem 5.2.4
thatZn^2 converges in distribution to aχ^2 (1) distribution.

Theorem 5.2.5(Slutsky’s Theorem).LetXn,X,An,andBnbe random variables

and letaandbbe constants. IfXn

D

→X,An

P

→a,andBn

P

→b,then

An+BnXn

D

→a+bX.

5.2.1 Bounded in Probability

Another useful concept, related to convergence in distribution, is boundedness in
probability of a sequence of random variables.
First consider any random variableXwith cdfFX(x). Then given>0, we
can boundXin the following way. Because the lower limit ofFXis 0 and its upper
limit is 1, we can findη 1 andη 2 such that

FX(x)</2forx≤η 1 andFX(x)> 1 −(/2) forx≥η 2.

Letη=max{|η 1 |,|η 2 |}.Then

P[|X|≤η]=FX(η)−FX(−η−0)≥ 1 −(/2)−(/2) = 1−. (5.2.7)

Thus random variables which are not bounded [e.g.,XisN(0,1)] are still bounded
in this probability way. This is a useful concept for sequences of random variables,
which we define next.

Definition 5.2.2(Bounded in Probability).We say that the sequence of random
variables{Xn}is bounded in probability if, for all> 0 , there exist a constant
B > 0 and an integerN such that

n≥N ⇒P[|Xn|≤B ]≥ 1 −.

Next, consider a sequence of random variables{Xn}which converges in distri-
bution to a random variableXthat has cdfF.Let>0 be given and chooseηso
that (5.2.7) holds forX. We can always chooseηso thatηand−ηare continuity
points ofF.Wethenhave

lim

n→∞

P[|Xn|≤η]≥ lim

n→∞

FXn(η)− lim

n→∞

FXn(−η−0) =FX(η)−FX(−η)≥ 1 −.

To be precise, we can then chooseNso large thatP[|Xn|≤η]≥ 1 − ,forn≥N.
We have thus proved the following theorem

Theorem 5.2.6.Let{Xn}be a sequence of random variables and letXbe a random
variable. IfXn→Xin distribution, then{Xn}is bounded in probability.