Page 350 - Elements of Distribution Theory

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336 Approximation of Probability Distributions

provided that

lim F n (x) = F(x)
n→∞
d
for all x ∈ R at which F is continuous.
Many of the properties of convergence in distribution, proven in this section for sequences
of real-valued random variables, extend to the case of random vectors. Several of these
extensions are presented below without proof; for further discussion and detailed proofs,
see, for example, Port (1994, Chapters 50 and 51).
The following result considers convergence in distribution of random vectors in terms
of convergence of expected values of bounded functions and generalizes Theorem 11.1 and
Corollaries 11.1 and 11.2.

Theorem 11.4. Let X 1 , X 2 ,... denote a sequence of d-dimensional random vectors and
let X denote a d-dimensional random vector.
(i) If
D
X n → X as n →∞
then

E[ f (X n )] → E[ f (X)] as n →∞
for all bounded, continuous, real-valued functions f .
(ii) If
E[ f (X n )] → E[ f (X)] as n →∞
for all bounded, uniformly continuous, real-valued functions f , then

D
X n → X as n →∞.
D D
(iii) If X n → Xas n →∞, and g is a continuous function, then g(X n ) → g(X).

Theorem 11.5 below generalizes Theorem 11.2 on the convergence of characteristic
functions.

Theorem 11.5. Let X 1 , X 2 ,... denote a sequence of d-dimensional random vectors and
let X denote a d-dimensional random vector.
Let ϕ n denote the characteristic function of X n and let ϕ denote the characteristic
function of X.

D
X n → X as n →∞
if and only if
d
lim ϕ n (t) = ϕ(t), for all t ∈ R .
n→∞
Recall that, when discussing the properties of characteristic functions of random vectors,
T
it was noted that two random vectors X and Y have the same distribution if and only if t X