Weak convergence in metric space

We give some definitions and basic properties of weak convergence of probability measures on metric spaces, based on [Bil99].

Let {(S, \mathcal{B}(S))} be a metric space. {\mathbb{P}_n, \mathbb{P}} are probability mesures on {(S, \mathbb{B}(S))}. We write {\mathbb{P} f = \int_S f d\mathbb{P}} for {\mathcal{B}(S)/\mathcal{B}(\mathbb{R})} measurable {f} for convenience.

Definition 1 {\mathbb{P}_n} is said to be weakly convergent to {\mathbb{P}}, denoted by {\mathbb{P}_n \Rightarrow \mathbb{P}}, if {\mathbb{P}_n f \rightarrow \mathbb{P} f} for every {f\in C_b(S; \mathbb{R})}.

The following theorem provides useful conditions equivalent to definition of weak convergence. A set {A\in \mathbb{B}(S)} is called {\mathbb{P}}-continuity set, if {\mathbb{P}(\partial A) = 0}.

Theorem 2 (The Portmanteau Theorem, [Bil99, Theorem 2.1]) These five conditions are equivalent:

  1. {\mathbb{P}_n \Rightarrow \mathbb{P}}.
  2. {\mathbb{P}_n f \rightarrow \mathbb{P} f} for all bounded uniformly continuous {f}.
  3. {\lim\sup_n \mathbb{P}_n F \le \mathbb{P} F} for all closed {F}
  4. {\lim\inf_n \mathbb{P}_n G \ge \mathbb{P} G} for all open {G}.
  5. {\mathbb{P}_n A \rightarrow \mathbb{P} A} for all {\mathbb{P}}-continuity sets {A}.

Let {\mathcal{P}} be the space of probability measures on {\mathcal{B}(S)}. {\mathbb{P}} can be metrized by Prohov distance {\pi(P, Q)} ([Bil99, P. 72]) if {S} is separable and complete metric space.

Theorem 3 ([Bil99, Theorem 6.8]) Suppose that {S} is separable and complete. Then

  1. weak convergence is equivalent to {\pi}-convergence,
  2. {\mathcal{P}} is separable and complete
  3. {\mathcal{P}} is relatively compact if and only if its {\pi}-closure is {\pi}-compact.

Let {(S', \mathcal{B}(S'))} be another metric space, and {h:S \rightarrow S'} is measurable {\mathcal{B}(S)/\mathcal{B}(S')}. Then {\mathbb{P}h^{-1}} is a probability measure on {\mathcal{B}(S')}.

Theorem 4 (Mapping theorem, [Bil99, Theorem 2.7]) Let {h:S \rightarrow S'} is measurable {\mathcal{B}(S)/\mathcal{B}(S')}, and {\mathbb{P}_n \Rightarrow \mathbb{P}}. In addition, {\mathbb{P} D_h = 0}, where {D_h \in \mathcal{B}(S)} be the set of discontinuities of {h}. Then {\mathbb{P}_n h^{-1} \Rightarrow \mathbb{P} h^{-1}}.

The above mapping theorem saids, weak convergence of probability measures are preserved by mapping {h}, if {h} is continuous almost surely with respect to the limit measure {\mathbb{P}}, i.e. {\mathbb{P}(D_h) = 0}.

Let {(\Omega, \mathcal{F}, \mathbb{P})} be a probability space, and {X:\Omega \rightarrow S} be measurable w.r.t. {\mathcal{F}/\mathcal{B}(S)}. Then, {X} is called random element. The distribution of {X} is the probability measure {\mathbb{P}X^{-1}} on {(\Omega, \mathcal{F})}, which is sometimes denoted by {\mathcal{L}(X)}. If {f: S \rightarrow \mathbb{R}} is a measurable {\mathcal{B}(S)/ \mathcal{B}(\mathbb{R}))}, then {\mathbb{E} [f(X)] = \mathbb{P}X^{-1}f}.

Definition 5 We say a sequence {\{X_n\}} converges in distribution to {X}, denoted by {X_n \Rightarrow X}, if {\mathbb{P}X_n^{-1} \Rightarrow \mathbb{P}X^{-1}}.

Thus, {X_n \Rightarrow X} if and only if {\mathbb{E}[f(X_n)] \rightarrow \mathbb{E}[f(X)]} for all {f\in C_b(S)}. We call a set {A \in \mathcal{B}(S)} as an {X}-continuity set if {\mathbb{P}(X\in \partial A) = 0}. Now, we can rewrite Theorem 2 in terms of random elements.

Theorem 6 ([Bil99, p. 26]) These five conditions are equivalent:

  1. {X_n \Rightarrow X}.
  2. {\mathbb{E}[ f(X_n)] \rightarrow \mathbb{E} [f(X)]} for all bounded uniformly continuous {f}.
  3. {\lim\sup_n \mathbb{P}(X_n \in F) \le \mathbb{P} (X\in F)} for all closed {F}
  4. {\lim\inf_n \mathbb{P} (X_n \in G) \ge \mathbb{P} (X\in G)} for all open {G}.
  5. {\mathbb{P} (X_n\in A) \rightarrow \mathbb{P} (X\in A)} for all {X}-continuity sets {A}.

Also, the Theorem 4 can be rewritten into mapping theorem of random elements: {X_n \Rightarrow X} on {S} implies {h(X_n) \Rightarrow h(X)} on {S'}, if {h} is continuous almost surely with respect to {X}, i.e. {\mathbb{P}(X\in D_h) = 0.}

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