Page 341 - Elements of Distribution Theory

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

11.2 Basic Properties of Convergence in Distribution 327

it follows that
M

dF n (x) = F n (M) − F n (−M) ≥ 1 − 2
−M
for sufficiently large n. Hence,
M
∞

f (x)[dF n (x) − dF(x)] + sup | f (x)|3 .

f (x)[dF n (x) − dF(x)] ≤
−∞ −M x
The same argument used in case 1 can be used to shown that
M

f (x)[dF n (x) − dF(x)] → 0as n →∞;

−M
since is arbitrary, it follows that
∞

f (x)[dF n (x) − dF(x)] → 0

−∞
as n →∞, proving the first part of the theorem.
Now suppose that E[ f (X n )] converges to E[ f (X)] for all real-valued, bounded, contin-
uous f . Define
1 if x < 0

h(x) = 1 − x if 0 ≤ x ≤ 1
0 if x > 1
and for any t > 0 define h t (x) = h(tx).
Note that, for fixed t, h t is a real-valued, bounded, continuous function. Hence, for all
t > 0,
lim E[h t (X n )] = E[h t (X)].
n→∞
For fixed x,

I {u≤x} ≤ h t (u − x) ≤ I {u≤x+1/t}
for all u, t. Hence,
∞

F n (x) ≤ h t (u − x) dF n (u) = E[h t (X n − x)]
−∞
and, for any value of t > 0,
lim sup F n (x) ≤ lim E[h t (X n − x)] = E[h t (X − x)] ≤ F(x + 1/t).
n→∞
n→∞
It follows that, if F is continuous at x,
lim sup F n (x) ≤ F(x). (11.1)
n→∞
Similarly, for fixed x,

I {u≤x−1/t} ≤ h t (u − x + 1/t) ≤ I {u≤x}
for all u, t > 0. Hence,
∞

F n (x) ≥ h t (u − x + 1/t) dF n (u) = E[h t (X n − x + 1/t)]
−∞