Page 338 - Elements of Distribution Theory

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324 Approximation of Probability Distributions

Of course, we expect that X 1 , X 2 ,... converges in distribution to a random variable
identically equal to 0; such a random variable has distribution function

0if x < 0
F(x) = .
1if x ≥ 0
Thus, lim n→∞ F n (x) = F(x)at all x = 0, that is, at all x at which F is continuous. Hence,
D
X n → 0as n →∞, where 0 may be viewed as the random variable equal to 0 with proba-
bility 1. However, if we require convergence of F n (x) for all x, X n would not have a limiting
distribution.

Often the random variables under consideration will require some type of standardization
in order to obtain a useful convergence result.

Example11.3 (Minimumofuniformrandomvariables). LetY 1 , Y 2 ,... denoteasequence
of independent, identically distributed random variables, each with a uniform distribution on
(0, 1). Suppose we are interested in approximating the distribution of T n = min(Y 1 ,..., Y n ).
For each n = 1, 2,..., T n has distribution function

H n (t) = Pr(T n ≤ t) = Pr{min(Y 1 ,..., Y n ) ≤ t}
0 if t < 0

= 1 − (1 − t) n if 0 ≤ t < 1.
1 if t ≥ 1
Fix t. Then
0if t ≤ 0

lim H n (t) =
n→∞ 1if t > 0
so that, as n →∞, T n converges in distribution to the random variable identically equal to
0. Hence, for any t > 0, Pr{min(Y 1 ,..., Y n ) ≤ t} can be approximated by 1. Clearly, this
approximation will not be very useful, or very accurate.
α
Now consider standardization of T n .For each n = 1, 2,..., let X n = n T n ≡
n min(Y 1 ,..., Y n ) where α is a given constant. Then, for each n = 1, 2,..., X n has distri-
bution function
α
F n (x; α) = Pr(X n ≤ x) = Pr{min(Y 1 ,..., Y n ) ≤ x/n }
0 if x < 0

α
α n
= 1 − (1 − x/n ) if 0 ≤ x < n .
1 if x ≥ n α
Fix x > 0. Then
1 if α< 1

lim F n (x; α) = 1 − exp(−x)if α = 1 .
n→∞
0 if α> 1
Thus, if α< 1, X n converges in distribution to the degenerate random variable 0, while
if α> 1, X n does not converge in distribution. However, if α = 1,

0 if x < 0
lim F n (x; α) =
n→∞ 1 − exp(−x)if 0 ≤ x < ∞,