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The Portmanteau Theorem

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.
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.
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.

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

1F≤f≤1Fε1_F \leq f \leq 1_{F^{\varepsilon}}
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.
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.
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.
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.
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, Inc., Hoboken, NJ, 2012