Theory of Statistics - George Mason University

86 1 Probability Theory
Theorem 1.34
Let {Fn} be a sequence of CDFs on IR. Let
and
Gn(x) = Fn(bgnx + agn)
Hn(x) = Fn(bhnx + ahn),
where {bdn} and {bhn} are sequences of positive real numbers and {agn} and
{ahn} are sequences of real numbers. Suppose
and
Gn
w
→ G
Hn w → H,
where G and H are nondegenerate CDFs. Then
and
bgn/bhn → b > 0,
(agn − ahn)/bgn → a ∈ IR,
H(bx + a) = G(x) ∀x ∈ IR.
Proof. ** fix
The distributions in Theorem 1.34 are in a location-scale family (see Section
2.6, beginning on page 178).
There are several necessary and sufficient conditions for convergence in
distribution. A set of such conditions is given in the following “portmanteau”
theorem.
Theorem 1.35 (characterizations of convergence in distribution;
“portmanteau” theorem)
Given the sequence of random variables Xn and the random variable X, all defined
on a common probability space, then each of the following is a necessary
d
and sufficient condition that Xn → X.
(i) E(g(Xn)) → E(g(X)) for all real bounded continuous functions g.
(ii) E(g(Xn)) → E(g(X)) for all real functions g such that g(x) → 0 as |x| →
∞.
(iii) Pr(Xn ∈ B) → Pr(X ∈ B) for all Borel sets B such that Pr(X ∈ ∂B) = 0.
(iv) lim inf Pr(Xn ∈ S) ≥ Pr(X ∈ S) for all open sets S.
(v) lim supPr(Xn ∈ T) ≤ Pr(X ∈ T) for all closed sets T.
Proof. The proofs of the various parts of this theorem are in Billingsley
(1995), among other resources.
Although convergence in distribution does not imply a.s. convergence, convergence
in distribution does allow us to construct an a.s. convergent sequence.
This is stated in Skorokhod’s representation theorem.
Theory of Statistics c○2000–2013 James E. Gentle