Page 351 - Elements of Distribution Theory

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052184472Xc11 CUNY148/Severini May 24, 2005 17:56

11.2 Basic Properties of Convergence in Distribution 337

T
and t Y have the same distribution for any vector t. Similarly, a very useful consequence of
Theorem 11.5 is the result that convergence in distribution of a sequence of d-dimensional
randomvectors X 1 , X 2 ,...toarandomvector X maybeestablishedbyshowingthat,forany
d
T
T
T
vector t ∈ R , t X 1 , t X 2 ,... converges in distribution to t X. Thus, a multidimensional
problem may be converted to a class of one-dimensional problems. This often called the
Cram´ er–Wold device.
Theorem 11.6. Let X, X 1 , X 2 ,... denote d-dimensional random vectors. Then
D
X n → X as n →∞
if and only if
D
T
T
t X n → t X as n →∞
d
for all t ∈ R .
Proof. Let ϕ n denote the characteristic function of X n and let ϕ denote the characteristic
T
function of X. Then t X n has characteristic function
˜ ϕ n (s) = ϕ n (st), s ∈ R

T
and t X has characteristic function
˜ ϕ(s) = ϕ(st), s ∈ R.

D
Suppose X n → X. Then
ϕ n (t) → ϕ(t) for all t ∈ R d

so that, fixing t,
ϕ n (st) → ϕ(st) for all s ∈ R,

D
T
T
proving that t X n → t X.
D
T
T
d
Now suppose that t X n → t X for all t ∈ R . Then
d
ϕ n (st) → ϕ(st) for all s ∈ R, t ∈ R .
Taking s = 1 shows that
d
ϕ n (t) → ϕ(t) for all t ∈ R ,
proving the result.
Thus, according to Theorem 11.6, convergence in distribution of the component random
variables of a random vector is a necessary, but not sufficient, condition for convergence in
distribution of the random vector. This is illustrated in the following example.

Example 11.10. Let Z 1 and Z 2 denote independent standard normal random variables
and, for n = 1, 2,..., let X n = Z 1 + α n Z 2 and Y n = Z 2 , where α 1 ,α 2 ,... is a sequence
D
of real numbers. Clearly, Y n → Z 2 as n →∞ and, if α n → α as n →∞, for some real