Page 343 - Elements of Distribution Theory

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

11.2 Basic Properties of Convergence in Distribution 329

Hence, for all x 1 , x 2 ,

|h(x 1 ) − h(x 2 )|≤|x 1 − x 2 |

so that, for a given value of t and a given > 0,
|h t (x 1 ) − h t (x 2 )|≤

whenever
|x 1 − x 2 |≤ /t.

It follows that h t is uniformly continuous and, hence, for all t > 0,
lim E[h t (X n )] = E[h t (X)].
n→∞
The proof of the corollary now follows as in the proof of Theorem 11.1.

The requirements in Theorem 11.1 that f is bounded and continuous are crucial for
the conclusion of the theorem. The following examples illustrate that convergence of the
expected values need not hold if these conditions are not satisfied.

Example 11.4 (Convergence of a sequence of degenerate random variables). As in
Example 11.2, let X 1 , X 2 ,... denote a sequence of random variables such that
Pr(X n = 1/n) = 1, n = 1, 2,....

D
We have seen that X n → 0as n →∞.
Let
0if x ≤ 0

f (x) = ;
1if x > 0
note that f is bounded, but it is discontinuous at x = 0. It is easy to see that E[ f (X n )] = 1
for all n = 1, 2,... so that lim n→∞ E[ f (X n )] = 1; however, E[ f (0)] = 0.

Example 11.5 (Pareto distribution). For each n = 1, 2,..., suppose that X n is a real-
valued random variable with an absolutely continuous distribution with density function
1 1 1
p n (x) = 1 1 , x > .
n(1 + n) n x 2+ n n + 1
Let F n denote the distribution function of X n ; then

1
0 if x < 1+n
F n (x) = −(1+ ) 1
1
1 − [(n + 1)x] n if ≤ x < ∞.
n+1
Hence,
0if x ≤ 0

lim F n (x) =
n→∞ 1if x > 0
D
so that X n → 0as n →∞.