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11.2 Basic Properties of Convergence in Distribution 325

which is the distribution function of a standard exponential distribution. Hence, if X is a
D
random variable with a standard exponential distribution, then n min(Y 1 ,..., Y n ) → X as
n →∞.
For instance, an approximation to

Pr{min(Y 1 ,..., Y 10 ) ≤ 1/10}= Pr{10 min(Y 1 ,..., Y 10 ) ≤ 1}
.
is given by 1 − exp(–1) = 0.632; the exact probability is 0.651.

The examples given above all have an important feature; in each case the distribution
functions F 1 , F 2 ,... are available. In this sense, the examples are not typical of those in
which convergence in distribution plays an important role. The usefulness of convergence
in distribution lies in the fact that the limiting distribution function may be determined in
cases in which the F n , n = 1, 2,..., are not available. Many examples of this type are given
in Chapters 12 and 13.

11.2 Basic Properties of Convergence in Distribution
Recall that there are several different ways in order to characterize the distribution of a
random variable. For instance, if two random variables X and Y have the same characteristic
function, or if E[ f (X)] = E[ f (Y)] for all bounded, continuous, real-valued functions f ,
then X and Y have the same distribution; see Corollary 3.1 and Theorem 1.11, respectively,
for formal statements of these results.
The results below show that these characterizations of a distribution can also be used to
characterize convergence in distribution. That is, convergence in distribution is equivalent
to convergence of expected values of bounded, continuous functions and is also equivalent
to convergence of characterstic functions. We first consider expectation.

Theorem 11.1. Let X 1 , X 2 ,... denote a sequence of real-valued random variables and
let X denote a real-valued random variable. Let X denote a set such that Pr(X n ∈ X) =
1, n = 1, 2,... and Pr(X ∈ X) = 1.

D
X n → X as n →∞
if and only if

E[ f (X n )] → E[ f (X)] as n →∞

for all bounded, continuous, real-valued functions f on X.

D
Proof. Suppose that X n → X as n →∞ and let F denote the distribution function of X.
In order to show that

E[ f (X n )] → E[ f (X)] as n →∞
for all bounded, continuous, real-valued functions f ,we consider two cases. In case 1, the
random variables X, X 1 , X 2 ,... are bounded; case 2 removes this restriction.