5.4.∗Extensions to Multivariate Distributions 353A result that will prove to be quite useful is the extension of the Δ-method; see
Theorem 5.2.9. A proof can be found in Chapter 3 of Serfling (1980).
Theorem 5.4.6.Let{Xn}be a sequence ofp-dimensional random vectors. Suppose
√
n(Xn−μ 0 )
D
→Np( 0 ,Σ).Letgbe a transformationg(x)=(g 1 (x),...,gk(x))′such that 1 ≤k≤pand the
k×pmatrix of partial derivatives,
B=[
∂gi
∂μj]
,i=1,...k;j=1,...,p ,are continuous and do not vanish in a neighborhood ofμ 0 .LetB 0 =Batμ 0 .Then
√
n(g(Xn)−g(μ 0 ))→DNk( 0 ,B 0 ΣB′ 0 ). (5.4.13)EXERCISES
5.4.1.Let{Xn}be a sequence ofp-dimensional random vectors. Show that
Xn
D
→Np(μ,Σ) if and only ifa′Xn
D
→N 1 (a′μ,a′Σa),for all vectorsa∈Rp.
5.4.2.LetX 1 ,...,Xnbe a random sample from a uniform(a, b) distribution. Let
Y 1 =minXiand letY 2 =maxXi. Show that (Y 1 ,Y 2 )′converges in probability to
the vector (a, b)′.
5.4.3.LetXnandYnbep-dimensional random vectors. Show that if
Xn−Yn
P
→ 0 andXn
D
→X,whereXis ap-dimensional random vector, thenYn→DX.
5.4.4.LetXnandYnbep-dimensional random vectors such thatXnandYnare
independent for eachnand their mgfs exist. Show that if
Xn→DXandYn→DY,whereXandYarep-dimensional random vectors, then (Xn,Yn)
D
→(X,Y).
5.4.5.SupposeXnhas aNp(μn,Σn) distribution. Show that
Xn
D
→Np(μ,Σ)iffμn→μandΣn→Σ.