Page 349 - Elements of Distribution Theory

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

11.2 Basic Properties of Convergence in Distribution 335

Since F(x j ) − F(x j−1 ) < /2 and F(x j−1 ) − F(x j ) > − /2, it follows that, for n ≥
max(N j , N j−1 ),
F n (x) − F(x) ≤
and

F n (x) − F(x) ≥− .
It follows that, for n ≥ max(N j , N j−1 ),

|F n (x) − F(x)| < .
Let N = max(N 0 , N 1 , N 2 ,..., N m+1 ). Then, for any value of x,
|F n (x) − F(x)| <

for n ≥ N; since the right-hand side of this inequality does not depend on x,it follows that,
given , there exists an N such that

sup |F n (x) − F(x)|≤
x
for n ≥ N, proving the result.

Example 11.9 (Convergence of F n (x n )). Suppose that a sequence of random variables
X 1 , X 2 ,... converges in distribution to a random variable X. Let F n denote the distribution
function of X n , n = 1, 2,..., and let F denote the distribution function of X, where F is
continuous on R. Then, for each x, lim n→∞ F n (x) = F(x).
Let x 1 , x 2 ,... denote a sequence of real numbers such that x = lim n→∞ x n exists. Then
|F n (x n ) − F(x)|≤|F n (x n ) − F(x n )|+|F(x n ) − F(x)|.

Since F is continuous,
lim |F(x n ) − F(x)|= 0;
n→∞
by Theorem 11.3,

lim |F n (x n ) − F(x n )|≤ lim sup |F n (x) − F(x)|= 0.
n→∞ n→∞ x
Hence,
lim |F n (x n ) − F(x)|= 0;
n→∞
that is, the sequence F n (x n ) converges to F(x)as n →∞.

Convergence in distribution of random vectors
We now consider convergence in distribution of random vectors. The basic definition is
a straightforward extension of the definition used for real-valued random variables. Let
X 1 , X 2 ,... denote a sequence of random vectors, each of dimension d, and let X denote
a random vector of dimension d.For each n = 1, 2,..., let F n denote the distribution
function of X n and let F denote the distribution function of X.We say that X n converges
in distribution to X as n →∞, written

D
X n → X as n →∞