Page 337 - Elements of Distribution Theory

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

11.1 Introduction 323

here θ 1 ,θ 2 ,... is a sequence of real numbers in the interval (0, 1). Let F n denote the
distribution function of X n . Then
0 if x < 0

F n (x) = θ n if 0 ≤ x < 1.
1 if x ≥ 1
Suppose that the sequence θ n , n = 1, 2,..., converges and let θ = lim n→∞ θ n . Then
0 if x < 0

lim F n (x) = F(x) ≡ θ if 0 ≤ x < 1.
n→∞
1 if x ≥ 1
Let X denote a random variable such that

Pr(X = 1) = 1 − Pr(X = 0) = θ.
D
Then X n → X as n →∞.
If the sequence θ n , n = 1, 2,..., does not have a limit, then X n , n = 1, 2,... does not
converge in distribution.

An important property of the definition of convergence in distribution is that we require
that the sequence F n (x), n = 1, 2,..., converges to F(x) only for those x that are continuity
points of F. Hence, if F is not continuous at x 0 , then the behavior of the sequence F n (x 0 ),
n = 1, 2,..., plays no role in convergence in distribution. The reason for this is that requir-
ing that F n (x), n = 1, 2,..., converges to F(x)at points at which F is discontinuous is too
strong of a requirement; this is illustrated in the following example.

Example 11.2 (Convergence of a sequence of degenerate random variables). Let
X 1 , X 2 ,... denote a sequence of random variables such that

Pr(X n = 1/n) = 1, n = 1, 2,....
Hence, when viewed as a deterministic sequence, X 1 , X 2 ,... has limit 0.
Let F n denote the distribution function of X n . Then

0if x < 1/n
F n (x) = .
1if x ≥ 1/n
Fix x. Clearly,
0if x < 0

lim F n (x) = .
n→∞ 1if x > 0
Consider the behavior of the sequence F n (0), n = 1, 2,.... Since 0 < 1/n for every
n = 1, 2,..., it follows that
lim F n (0) = 0
n→∞
so that
0if x ≤ 0

lim F n (x) = G(x) ≡ .
n→∞ 1if x > 0
Note that, since G is not right-continuous, it is not the distribution function of any random
variable.