Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

5.2. Convergence in Distribution 335

Thelittle-onotation is used in terms of convergence in probability, also. We
often writeop(Xn), which means

Yn=op(Xn) if and only ifXYnn

P

→0, asn→∞. (5.2.10)

There is a correspondingbig-Opnotation, which is given by

Yn=Op(Xn) if and only ifXYnnis bounded in probability asn→∞. (5.2.11)

The following theorem illustrates the little-o notation, but it also serves as a
lemma for Theorem 5.2.9.

Theorem 5.2.8.Suppose{Yn}is a sequence of random variables that is bounded

in probability. SupposeXn=op(Yn).ThenXn
P
→ 0 ,asn→∞.

Proof: Let>0 be given. Because the sequence{Yn}is bounded in probability,
there exist positive constantsN andB such that

n≥N =⇒P[|Yn|≤B ]≥ 1 −. (5.2.12)

Also, becauseXn=op(Yn), we have

Xn

Yn

P

→ 0 , (5.2.13)

asn→∞.Wethenhave

P[|Xn|≥ ]=P[|Xn|≥,|Yn|≤B ]+P[|Xn|≥,|Yn|>B ]

≤ P

[

Xn

|Yn|

≥

B

]

+P[|Yn|>B ].

By (5.2.13) and (5.2.12), respectively, the first and second terms on the right side
can be made arbitrarily small by choosingnsufficiently large. Hence the result is
true.
We can now prove the theorem about the asymptotic procedure, which is often
called the Δ method.

Theorem 5.2.9.Let{Xn}be a sequence of random variables such that

√

n(Xn−θ)

D

→N(0,σ^2 ). (5.2.14)

Suppose the functiong(x)is differentiable atθandg′(θ) =0.Then

√

n(g(Xn)−g(θ))→DN(0,σ^2 (g′(θ))^2 ). (5.2.15)

Proof:Using expression (5.2.9), we have

g(Xn)=g(θ)+g′(θ)(Xn−θ)+op(|Xn−θ|),