Page 354 - Elements of Distribution Theory

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

340 Approximation of Probability Distributions

then
Pr(|X n − X|≥ 1/2) = Pr(X n = 1 ∩ X = 0) + Pr(X n = 0 ∩ X = 1)
1 n + 1 1 n − 1
= +
4 n 4 n
1
= ;
2
it follows that X n does not converge to X in probability.
On the other hand, suppose that
1
Pr(X n = 1|X = 1) = 1 and Pr(X n = 1|X = 0) = .
n
Note that
Pr(X n = 1) = Pr(X n = 1|X = 1)Pr(X = 1) + Pr(X n = 1|X = 0)Pr(X = 0)
1 1 1
= +
2 n 2
1 n + 1
= ,
2 n
as stated above. In this case, for any > 0,
1
Pr(|X n − X|≥ ) = Pr(X n = 1 ∩ X = 0) + Pr(X n = 0 ∩ X = 1) =
2n
p
so that X n → X as n →∞.

The preceding example shows that convergence in distribution does not necessarily imply
convergence in probability. The following result shows that convergence in probability does
imply convergence in distribution. Furthermore, when the limiting random variable is a
constant with probability 1, then convergence in probability is equivalent to convergence in
distribution.

Corollary 11.3. Let X, X 1 , X 2 ,... denote real-valued random variables.
p D
(i) If X n → Xas n →∞ then X n → Xas n →∞.
D p
(ii) If X n → Xas n →∞ and Pr(X = c) = 1 for some constant c, then X n → Xas
n →∞.

p
Proof. Suppose that X n → X. Consider the event X n ≤ x for some real-valued x.If
X n ≤ x, then, for every > 0, either X ≤ x + or |X n − X| > . Hence,
Pr(X n ≤ x) ≤ Pr(X ≤ x + ) + Pr(|X n − X| > ).
Let F n denote the distribution function of X n and let F denote the distribution function of
F. Then, for all > 0,
lim sup F n (x) ≤ F(x + ).
n→∞
Similarly, if, for some > 0, X ≤ x − , then either X n ≤ x or |X n − X| > . Hence,
F(x − ) ≤ F n (x) + Pr(|X n − X| > )