0.1 Portmanteau Theorem
In mathematics and statistics, weak convergence is one of many types of convergence relating to the convergence in probability. It depends on a topology on the underlying space and thus is not a purely measure-theoretic notion.
There are several equivalent definitions of weak convergence of a sequence of measures, some of which are (apparently) more general than others. The equivalence of these conditions is sometimes known as the Portmanteau theorem.
The following statements are equivalent: 1. \(X_{n}\stackrel{\mathcal{D}}{\to}X\); 2. \(\mathbb{E}[g(X_{n})]\to \mathbb{E}[g(X)]\) for all bounded and continuous functions \(g\); 3. \(\mathbb{E}[g(X_{n})]\to \mathbb{E}[g(X)]\) for all \(g\) of the form \(g(x)=f(x)\mathbb{1}_{[a,b]}(x)\), where \(f\) is continuous on \([a,b]\) and \(a,b \in C(F)\), that is, \(g\) is a continuous function with finite support \([a,b]\) with \(a,b \in C(F)\); 4. \(\mathbb{E}[g(X_{n})]\to \mathbb{E}[g(X)]\) for all \(g\) bounded and uniformly continuous; 5. If \(A\) is such that \(\mathbb{P}(X \in \delta A)=0\) then \(\mathbb{P}(X_{n}\in A)\to \mathbb{P}(X\in A).\)
- In (5) \(\delta A\) denotes the boundary of a set \(A\). These are points that are reachable both form inside the set and outside: \[ \delta A=\{ x : \exists \{ y_{n} \}\in A~s.t. y_{n}\to x~\&~ \exists \{ z_{n} \}\in A^c~s.t.~z_{n} \to x \}. \]
- Equivalent formulation of (5) using d.f.s \(F_{n}\) and \(F\): \[ F(\delta A)=0\implies F_{n}(A)\to F(A). \] Here \(F((a,b]):=F(b)-F(a)=\int \mathbb{1}_{(a,b]}(x)dF(x)\).