Page 340 - Elements of Distribution Theory

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

326 Approximation of Probability Distributions

Case 1: Suppose that there exists a constant M such that, with probability 1,

|X n |≤ M, n = 1, 2,...
and |X|≤ M. We may assume, without loss of generality, that M is a continuity point of F.
Consider a bounded, continuous, function f : R → R and let > 0. Let x 1 , x 2 ,..., x m
denote continuity points of F such that

−M = x 0 < x 1 < ··· < x m−1 < x m < x m+1 = M
and

max sup | f (x) − f (x i )|≤ .
1≤i≤m x i ≤x≤x i+1
Define

f m (x) = f (x i ) for x i ≤ x < x i+1 .
Then, as n →∞,
M m

f m (x) dF n (x) = f (x i )[F n (x i+1 ) − F n (x i )]
−M i=1
m

→ f (x i )[F(x i+1 ) − F(x i )]
i=1
M
= f m (x) dF(x).
−M
Hence, for sufficiently large n,

M

f m (x)[dF n (x) − dF(x)] ≤ .

−M
It follows that
M

f (x)[dF n (x) − dF(x)]
−M
M M

≤ [ f (x) − f m (x)][dF n (x) − dF(x)] + f m (x)[dF n (x) − dF(x)]
−M −M
≤ 3 .
Since is arbitrary, it follows that
lim E[ f (X n )] = E[ f (X)].
n→∞
Case 2: For the general case in which the X 1 , X 2 ,... and X are not necessarily bounded,
let 0 < < 1be arbitrary and let M and −M denote continuity points of F such that

M
dF(x) = F(M) − F(−M) ≥ 1 − .
−M
Since

F n (M) − F n (−M) → F(M) − F(−M)as n →∞,