Page 345 - Elements of Distribution Theory

P1: JZP
052184472Xc11 CUNY148/Severini May 24, 2005 17:56

11.2 Basic Properties of Convergence in Distribution 331

Proof. Applying Theorem 11.1 to the real and imaginary parts of exp{itx} shows that
D
X n → X implies that ϕ n (t) → ϕ(t) for all t.
Hence, suppose that
lim ϕ n (t) = ϕ(t), for all −∞ < t < ∞.
n→∞
Let g denote an arbitrary bounded uniformly continuous function such that

|g(x)|≤ M, −∞ < x < ∞.
If it can be shown that

lim E[g(X n )] = E[g(X)] as n →∞,
n→∞
D
then, by Corollary 11.1, X n → X as n →∞ and the theorem follows.
Given > 0, choose δ so that
sup |g(x) − g(y)| < .
x,y:|x−y|<δ
Let Z denote a standard normal random variable, independent of X, X 1 , X 2 ,..., and con-
sider |g(X n + Z/t) − g(X n )| where t > 0. Whenever |Z|/t is less than δ,

|g(X n + Z/t) − g(X n )| < ;
whenever |Z|/t ≥ ,we still have

|g(X n + Z/t) − g(X n )|≤|g(X n + Z/t)|+|g(X n )|≤ 2M.
Hence, for any t > 0,

E[|g(X n + Z/t) − g(X n )|] ≤ Pr(|Z| < tδ) + 2MPr(|Z| > tδ).
For sufficiently large t,

Pr(|Z| > tδ) ≤
2M
so that

E[|g(X n + Z/t) − g(X n )|] ≤ 2 .
Similarly, for sufficiently large t,

E[|g(X + Z/t) − g(X)|] ≤ 2
so that
E[|g(X n ) − g(X)|] ≤ E[|g(X n ) − g(X n + Z/t)|] + E[|g(X n + Z/t) − g(X + Z/t)|]
+ E[|g(X + Z/t) − g(X)|] ≤ 4 + E[|g(X n + Z/t) − g(X + Z/t)|].

Recall that, by Example 3.2,
1
ϕ Z (t) = (2π) 2 φ(t),
where φ denotes the standard normal density.