Theorem 10.1.1
(Portmanteau Theorem)
Let µ, (µn) be probability measures on (R, B(R)). The following are
equivalent:
R
R
1.
R f dµn → R f dµ for all bounded, continuous f : R → R.
2.
µn(A) → µ(A) for all A ∈ B(R) with µ(∂A) = 0.
3.
Fn(x) → F(x) for all x ∈ R where F is continuous at x.
4. (Skorohod’s theorem) There are random variables Y, Y1, Y2, . . .
defined on some probability space, with LY = µ and LYn = µn
for each n ≥ 1, so that Yn → Y almost surely.
R
R
f
dµ
→
5.
n
R f dµ for all bounded, Borel measurable f :
R
R → R with µ(Df) = 0, where Df is the set of discontinuity
points of f.
Proving Theorem 10.1.1
(1)
(3)
(2)
(4)
(5)
Examples:
1. (Delta measures) µn = δxn
2. (Gaussian measures) µn = N(µn, σ2n)
3. (Approximating Lebesgue measure) µn =
1
n
Pn
i=1 δi/n
4. (Random measures?) Let P(R) be the space of all probability measures on (R, B(R)). Now let X1, X2, X3, . . . be i.i.d. random variables
on (Ω, F, P) with law L(Xi) = µ. Define the random measure
1X
Φ(ω) =
δXi(ω) ∈ P(R).
n
n
i=1
Exercises:
10.3.1. Suppose that L(Xn) ⇒ δc for some c ∈ R. Prove that {Xn}
converges to c in probability.
10.3.2. Let X, Y1, Y2, . . . be independent random variables, with
P(Yn = 1) = 1/n and P(Yn = 0) = 1 − 1/n. Let Zn = X + Yn.
Prove that L(Zn) ⇒ L(X), i.e., that the law of Zn converges to the
law of X.
10.3.3. Let µn = N(0, 1/n) be a normal distribution with mean
zero and variance 1/n. Does the sequence {µn} converge weakly to
some probability measure? If yes, to what measure?
10.3.4. Prove that weak limits, if they exist, are unique. That is,
if µ, ν, µ1, µ2, . . . are probability measures, and µn ⇒ µ and also
µn ⇒ ν, then µ = ν.