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The Portmanteau Theorem

It is important to consider criteria of weak convergence not only on Euclidean spaces, but also on any metric space (S,S)(S, \mathcal{S}). The Portmanteau theorem provides several equivalent formulations of weak convergence, and any of them could serve as the definition.
PORTMANTEAU THEOREM
Let X,X1,X2,X3,…X, X_1, X_2, X_3, \dots be random elements in metric space (S,S)(S, \mathcal{S}). Then these four conditions are equivalent:
  • Xn→dXX_n \xrightarrow{d} X.
  • lim sup⁡n→∞P{Xn∈F}≤P{X∈F}\limsup_{n\to\infty} \mathbb{P}\{ X_n \in F \} \leq \mathbb{P}\{ X \in F \} for all closed FF.
  • lim inf⁡n→∞P{Xn∈G}≥P{X∈G}\liminf_{n\to\infty} \mathbb{P}\{ X_n \in G \} \geq \mathbb{P}\{ X \in G \} for all open GG.
  • P{Xn∈A}→P{X∈A}\mathbb{P}\{ X_n \in A \} \rightarrow \mathbb{P}\{ X \in A \} for all P\mathbb{P}-continuity sets AA.

What is the P\mathbb{P}-continuity set?

Interior, boundary, and exterior of a set $A$
Interior, boundary, and exterior of a set AA
Before defining a P\mathbb{P}-continuity set, let us recall the basic topological pieces of a set. We say that xx is an interior point of AA if there is an open ball B(x,r)B(x,r) centered at xx with radius r>0r>0 such that B(x,r)⊂AB(x,r)\subset A.
INTERIOR
The interior of AA is
A∘:={x∈S:∃r>0 such that B(x,r)⊂A}.\begin{equation*}A^{\circ}:=\{x\in S: \exists r>0\text{ such that }B(x,r)\subset A\}.\end{equation*}
We say that a point xx belongs to the closure of AA if every open ball B(x,r)B(x,r) centered at xx, no matter how small the radius r>0r>0 is, contains at least one point from AA.
CLOSURE AND BOUNDARY
The closure of AA is
A‾:={x∈S: for every r>0, B(x,r)∩A≠∅}.\begin{equation*}\overline{A}:=\{x\in S:\text{ for every }r>0,\ B(x,r)\cap A\neq\varnothing\}.\end{equation*}
Its boundary is defined by
∂A:=A‾∖A∘,\begin{equation*}\partial A := \overline{A} \setminus A^{\circ},\end{equation*}
The boundary ∂A\partial A of AA consists of the points where every small ball sees both AA and its complement. Equivalently, ∂A\partial A is the closure of AA minus its interior.
P\mathbb{P}-continuity set
Let A∈SA \in \mathcal{S}. We call AA a P\mathbb{P}-continuity set if
P(X∈∂A)=0.\begin{equation*}\mathbb{P}(X \in \partial A)=0.\end{equation*}
In words, AA is a P\mathbb{P}-continuity set when the limiting random element XX has zero probability of landing exactly on the boundary of AA.

Proof of Portmanteau Theorem

The following proof is partially taken from Billingsley [Billingsley2012], Theorem 25.8.

1 ⇒\Rightarrow 2

Assume 1 and fix a closed set FF. Introduce the approximation of the indicator 1F\mathbf{1}_{F} by the function
f(x)=max⁡(0,1−ρ(x,F)ε).\begin{equation*}f(x) = \max\left(0, 1 - \frac{\rho(x, F)}{\varepsilon}\right).\end{equation*}
It is bounded and continuous, so it can be used in the definition of weak convergence. Moreover,
1F(x)≤f(x)=max⁡(0,1−ρ(x,F)ε)≤1Fε(x)\begin{equation*}\mathbf{1}_{F}(x) \leq f(x) = \max\left(0, 1 - \frac{\rho(x, F)}{\varepsilon}\right) \leq \mathbf{1}_{F^{\varepsilon}}(x)\end{equation*}
Therefore,
lim sup⁡n→∞E1F(Xn)≤lim sup⁡n→∞Ef(Xn)=Ef(X)≤E1Fε(X)=P{X∈Fε}\begin{equation*}\limsup_{n\to\infty} \mathbb{E} \mathbf{1}_{F} (X_n) \leq \limsup_{n\to\infty} \mathbb{E} f(X_n) = \mathbb{E} f(X) \leq \mathbb{E} \mathbf{1}_{F^{\varepsilon}}(X) = \mathbb{P}\{ X \in F^{\varepsilon} \}\end{equation*}
Letting ε↓0\varepsilon \downarrow 0 gives 2.

2 ⇒\Rightarrow 3

Let GG be open and set F:=GcF := G^c, which is closed. Then
P{Xn∈G}=1−P{Xn∈F},P{X∈G}=1−P{X∈F}.\begin{equation*}\mathbb{P}\{X_n \in G\} = 1 - \mathbb{P}\{X_n \in F\},\qquad\mathbb{P}\{X \in G\} = 1 - \mathbb{P}\{X \in F\}.\end{equation*}
Using 2 for the closed set FF, we obtain
lim inf⁡n→∞P{Xn∈G}=1−lim sup⁡n→∞P{Xn∈F}≥1−P{X∈F}=P{X∈G}.\begin{equation*}\liminf_{n\to\infty} \mathbb{P}\{X_n \in G\}=1 - \limsup_{n\to\infty} \mathbb{P}\{X_n \in F\}\geq1 - \mathbb{P}\{X \in F\}=\mathbb{P}\{X \in G\}.\end{equation*}
Thus 3 holds.

3 ⇒\Rightarrow 2

Let FF be closed and set G:=FcG := F^c, which is open. Then
P{Xn∈F}=1−P{Xn∈G},P{X∈F}=1−P{X∈G}.\begin{equation*}\mathbb{P}\{X_n \in F\} = 1 - \mathbb{P}\{X_n \in G\},\qquad\mathbb{P}\{X \in F\} = 1 - \mathbb{P}\{X \in G\}.\end{equation*}
Using 3 for the open set GG, we obtain
lim sup⁡n→∞P{Xn∈F}=1−lim inf⁡n→∞P{Xn∈G}≤1−P{X∈G}=P{X∈F}.\begin{equation*}\limsup_{n\to\infty} \mathbb{P}\{X_n \in F\}=1 - \liminf_{n\to\infty} \mathbb{P}\{X_n \in G\}\leq1 - \mathbb{P}\{X \in G\}=\mathbb{P}\{X \in F\}.\end{equation*}
Thus 2 holds.

2 and 3 ⇒\Rightarrow 4

If A∘A^{\circ} and Aˉ\bar{A} are the interior and closure of AA, then 2 and 3 together imply
P{X∈Aˉ}≥2lim sup⁡n→∞P{Xn∈Aˉ}≥A⊂Aˉlim sup⁡n→∞P{Xn∈A}≥lim inf⁡n→∞P{Xn∈A}≥A∘⊂Alim inf⁡n→∞P{Xn∈A∘}≥3P{X∈A∘}.\begin{equation*}\mathbb{P} \{ X \in \bar{A} \} \overset{2}{\geq} \limsup_{n\to\infty} \mathbb{P} \{ X_n \in \bar{A} \} \overset{A\subset \bar{A}}{\geq} \limsup_{n\to\infty} \mathbb{P} \{ X_n \in A \} \geq \liminf_{n\to\infty} \mathbb{P} \{ X_n \in A \} \overset{A^{\circ}\subset A}{\geq} \liminf_{n\to\infty} \mathbb{P} \{ X_n \in A^{\circ} \} \overset{3}{\geq} \mathbb{P} \{ X \in A^{\circ} \} .\end{equation*}
If AA is a P\mathbb{P}-continuity set, then the extreme terms both equal P(A)\mathbb{P}(A), so all inequalities above are equalities. Hence 4 follows.

4 ⇒\Rightarrow 1

To prove 1, it is enough to show that for every bounded continuous function ff. By shifting and scaling ff, we may assume 0<f<10<f<1. Then
Ef(X)=∫0∞P{f(X)>t}dt=∫01P{f(X)>t}dt.\begin{equation*}\mathbb{E} f(X) =\int_0^{\infty} \mathbb{P} \{ f(X)>t \} \textrm{d} t=\int_0^1 \mathbb{P} \{ f(X)>t \} \textrm{d} t.\end{equation*}
For each t∈(0,1)t\in(0,1), set
At:={x∈S:f(x)>t}.\begin{equation*}A_t:=\{x\in S:f(x)>t\}.\end{equation*}
Since ff is continuous,
∂At⊂{x∈S:f(x)=t}.\begin{equation*}\partial A_t \subset \{x\in S:f(x)=t\}.\end{equation*}
Hence AtA_t is a P\mathbb{P}-continuity set whenever
P(f(X)=t)=0.\begin{equation*}\mathbb{P}(f(X)=t)=0.\end{equation*}
There are at most countably many such exceptional values of tt. By 4 and the bounded convergence theorem,
Ef(Xn)=∫01P{f(Xn)>t}dt→∫01P{f(X)>t}dt=Ef(X)\begin{equation*}\mathbb{E} f(X_n) = \int_0^1 \mathbb{P} \{ f(X_n) > t \} \textrm{d} t \rightarrow \int_0^1 \mathbb{P} \{ f(X) > t \} \textrm{d} t = \mathbb{E} f(X)\end{equation*}
which proves 1. Therefore the four conditions are equivalent. □\Box

References

  • [Billingsley2012]Billingsley, Patrick. Probability and Measure. Anniversary edition. John Wiley & Sons, 2012.