Portmanteau Theorem

We recall definition of convergence in distribution and its related Portmanteau Theorem here. This is based on [Bil99] ([Patrick Billingsley 1999]).

Definition 1

{X} is called random element if it satisfies the following conditions:

  1. {(\Omega, \mathcal F, \mathbb P)} is a probability space;
  2. {X: \Omega \mapsto S} for some metric space {S}.

Random elements {X_{n}} are said to be convergent to an random element {X} in distribution, and denoted by {X_{n} \Rightarrow X}, if

\displaystyle  \lim_{n\rightarrow \infty} \mathbb E[ f(X_{n}) ] = \mathbb E[ f(X) ], \quad \forall f\in C_{b}(S).

– QED. –

The following theorem can be found in Page 26 of [Bil99].

Theorem 2 (Portmanteau Theorem)

The following five statements 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) = \mathbb P(X\in A)} for all {X}-continuity sets {A}.

– QED –

No. 3 of Pormanteau Theorem can be written by

\displaystyle \lim\sup_{n} \mathbb E I_{F}(X_{n}) \le \mathbb E I_{F}(X),

which is indeed to be still true for all bounded upper semicontinuous functions in place of {I_{F}}. Similarly, No. 4 is equivalent to {\lim\inf_{n} \mathbb E I_{G}(X_{n}) \ge \mathbb E I_{G}(X),} which can be extended to all bounded lower semicontinuous functions.

Example 1

If {X_{n} \Rightarrow X}, then

  1. {\lim\sup_{n} \mathbb E f(X_{n}) \le \mathbb E f(X)} for all bounded upper semicontinuous {f};
  2. {\lim\inf_{n} \mathbb E f(X_{n}) \ge \mathbb E f(X)} for all bounded lower semicontinuous {f}.

Proof: If {f} is bounded upper semicontinuous, then there exists bounded continuous functions {f_{\epsilon}} such that {f_{\epsilon} \downarrow f}. Thus,

\displaystyle \lim\sup_{n\rightarrow \infty} \mathbb E [f(X_{n})] \le \lim\sup_{n} \mathbb E[ f_{\epsilon}(X) ] = \mathbb E [f_{\epsilon}(X)].

Note that, { \mathbb E [f_{\epsilon}(X)] \downarrow \mathbb E [f(X)]} by bounded convergence theorem, and it yields the first conclusion. The second one is the consequence of the first conclusion applying to upper semicontinuous function {(-f)}. \Box

– QED –

We can also extend the above example even further to {\mathbb PX^{-1}}-almost semicontinuous functions.

Example 2

Let {X_{n} \Rightarrow X}.

  1. If {f} is bounded upper semicontinuous {\mathbb PX^{-1}}-almost surely, then {\lim\sup_{n} \mathbb E f(X_{n}) \le \mathbb E f(X)};
  2. If {f} is bounded lower semicontinuous {\mathbb PX^{-1}}-almost surely, then {\lim\inf_{n} \mathbb E f(X_{n}) \ge \mathbb E f(X)}.

Proof: We will only show the first result. The condition

{f} is upper semicontinuous {\mathbb PX^{-1}}-almost surely

means that

{\exists N \in \mathcal B(S)} such that {\mathbb P(X\in N) = 0} and {f} is USC in {S\setminus N}.

Denote {K = \sup_{S} |f(x)| <\infty}, and define {g = f (1 - I_{N}) + K I_{N}}. Then {g} is bounded USC and this yields by Example 1

\displaystyle \lim\sup_{n} \mathbb E g(X_{n}) \le \mathbb E g(X).

One can conclude the result by easily checking that

\displaystyle \mathbb E g(X) = \mathbb E f(X), \hbox{ and } \mathbb E g(X_{n}) \ge \mathbb E f(X_{n}).

\Box

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