Page 344 - Elements of Distribution Theory

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330 Approximation of Probability Distributions

Consider the function f (x) = x, which is continuous, but unbounded. It is straight
forward to show that
1 ∞ 1
E[ f (X n )] = dx = 1, n = 1, 2,...
1 1
n(1 + n) n 1 x 1+ n
n+1
while E[ f (0)] = 0.

A useful consequence of Theorem 11.1 is the result that convergence in distribution is
preserved under continuous transformations. This result is stated in the following corollary;
the proof is left as an exercise.

Corollary 11.2. Let X, X 1 , X 2 ,... denote real-valued random variables such that

D
X n → X as n →∞.
Let f : X → R denote a continuous function, where X ⊂ R satisfies Pr[X n ∈ X] = 1,
n = 1, 2,... and Pr(X ∈ X) = 1. Then
D
f (X n ) → f (X) as n →∞.

Example 11.6 (Minimum of uniform random variables). As in Example 11.3, let
Y 1 , Y 2 ,... denote a sequence of independent identically distributed, each with a uniform
distribution on (0, 1) and let X n = n min(Y 1 ,..., Y n ), n = 1, 2,.... In Example 11.3, it
D
was shown that that X n → X as n →∞, where X is a random variable with a standard
exponential distribution.
Let W n = exp(X n ), n = 1, 2,.... Since exp(·)isa continuous function, it follows from
D
Corollary 11.2 that W n → W as n →∞, where W = exp(X). It is straightforward to show
that W has an absolutely continuous distribution with density
1
, w ≥ 1.
w 2
An important result is that convergence in distribution may be characterized in terms of
convergence of characteristic functions. The usefulness of this result is due to the fact that
the characteristic function of a sum of independent random variables is easily determined
from the characteristic functions of the random variables making up the sum. This approach
is illustrated in detail in Chapter 12.

Theorem 11.2. Let X 1 , X 2 ,... denote a sequence of real-valued random variables and
let X denote a real-valued random variable. For each n = 1, 2,..., let ϕ n denote the
characteristic function of X n and let ϕ denote the characteristic function of X. Then

D
X n → X as n →∞
if and only if

lim ϕ n (t) = ϕ(t), for all −∞ < t < ∞.
n→∞