Weak Convergence

• $C_b(S)$,
  • the set of boubded continuous functions on $S$.

Definition 1 Weak convergence

• $(S,\rho )$,
  • metric space with Borel $\sigma$-algebra $\mathcal{B}(S))$,
• $\{P_n{\}}_{n\in \mathbb{N}}$,
  • a sequence of probability measure on $(S,\mathcal{B}(S))$,
• $P$
  • a measure on $(S,\mathcal{B}(S))$

$\{P_n\}$ is said to converge wealkly to $P$ if

$$\mathrm{\forall }f\in C_b(S),\underset{n\to \mathrm{\infty }}{lim}{\int }_{S}f(s)P_n(ds)={\int }_{S}f(s)P(ds).$$

We write $P_n\stackrel{w}{\to }P$.

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Proposition 1

• $(S,\rho )$,
• $\mathcal{O}$,
  • all open sets in $S$,

(1)

If $P_n\stackrel{w}{\to }P$, $P_n\stackrel{w}{\to }{P}^{\prime }$, then $P=P^{\prime }$.

(2) Let $F$ be closed set.

$$\begin{array}{}\text{(1)}& f(x)& :=& (1-\rho (x,F)/ϵ{)}^{+}\end{array}$$

(2-1) $f$ is bounded.

(2-2) $f$ is uniformly continuous

(2-3)

$$\begin{array}{}\text{(2)}& I_F(x)\le f(x)\le I_{F^{ϵ}}(x).\end{array}$$

proof

proof of (1)

$$\begin{array}{rcl}\mathrm{\forall }O\in \mathcal{O},P(O)& =& {\int }_S{1}_O(x)P(dx)\\ & =& \underset{n\to \mathrm{\infty }}{lim}{\int }_S{1}_O(x)P_n(dx)\\ & =& {\int }_S{1}_O(x)P^{\prime }(dx)\\ & =& P^{\prime }(O)\end{array}$$

proof of (2)

Since $0\le f_{ϵ}\le 1$, (2-1) holds.

As to (2-2),

$$\begin{array}{}\text{(3)}& |f_{ϵ}(x)-f_{ϵ}(x)|\le \end{array}$$

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Definition 3

• $(S,\rho )$,
  • metric space
• $\{({\mathrm{\Omega }}_n,{\mathcal{F}}_n,P_n){\}}_{n\in \mathbb{N}}$,
  • a sequence of probability spaces
• $X_n:\mathrm{\Omega }\to S$,
  • random variable over $({\mathrm{\Omega }}_n,{\mathcal{F}}_n,P_n)$,
• $(\mathrm{\Omega },\mathcal{F},P)$,
  • probability space
  • random variable on $({\mathrm{\Omega }}_n,{\mathcal{F}}_n,P_n)$,
• $X:\mathrm{\Omega }\to S$,
  • random variable over $(\mathrm{\Omega },\mathcal{F},P)$,

$\{X_n\}$ is said to converge to $X$ in distribution if

$$\begin{array}{rcl}& & P_n{X}_n^{-1}\stackrel{w}{\to }P{X}^{-1}\\ \text{(4)}& & \iff & \mathrm{\forall }f\in C_b(S),\underset{n\to \mathrm{\infty }}{lim}{\int }_{S}f(s)(P_n{X}_n^{-1})(ds)={\int }_{S}f(s)(P{X}^{-1})(ds).\end{array}.$$

We write $X_n\stackrel{d}{\to }X$.

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Proposition 4

(1)

$X_n\stackrel{d}{\to }X$ if and only if

$$\mathrm{\forall }f\in C_b(S),\underset{n\to \mathrm{\infty }}{lim}{\int }_{S_n}f(X_n(\omega ))P_n(d\omega )={\int }_{S}f(X(\omega ))P(d\omega ).$$

proof

$◻$

Definition 5

• $(S,\rho )$,
  • metric space
• $A\in \mathcal{B}(S)$

$A$ is said to be $P$-continuity set if

$$P(\partial A)=0$$

where $\partial A$ is boundary of $A$. Note that $\partial A$ is closed.

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Theorem 6 The Portmanteau Theorem

• $(S,\rho )$,
  • metric space
• $\mathcal{O}$,
  • all open sets
• $\mathcal{F}$,
  • all closed sets

These five conditions are equivalent:

• (i) $P_n\stackrel{w}{\to }P$,
• (ii) $P_{n}f\to Pf$ for all bounded, uniformly continuous $f$,
• (iii) $lim\underset{n}{sup}{P}_{n}F\le PF$ for all $F\in \mathcal{F}$
• (iv) $lim\underset{n}{inf}{P}_{n}G\ge PG$ for all $G\in \mathcal{O}$
• (v) $P_{n}A\to PA$ for all $P$ continuity sets $A$

proof

(i) $\Rightarrow$ (ii)

(ii) $\Rightarrow$ (iii)

(iii) $\iff$ (iv)

(iii) $\Rightarrow$ (v)

(v) $\Rightarrow$ (i)

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Theorem 7

• $(\mathrm{\Omega },\mathcal{F},P)$,
• $X:\mathrm{\Omega }\to C[0,\mathrm{\infty })$,
  • $\mathcal{F}/\mathcal{B}(C[0,\mathrm{\infty }))$-measurable
• $({\mathrm{\Omega }}_n,{\mathcal{F}}_n,P_n)$,
  • probability spaces
• $X^n:{\mathrm{\Omega }}_n\to C[0,\mathrm{\infty })$,
  • ${\mathcal{F}}_n/\mathcal{B}(C[0,\mathrm{\infty }))$-measurable

If

$$X^n\stackrel{d}{\to }X,$$

then for all $\tilde{t}:=(t_1,\dots ,t_d)\in \mathcal{T}$,

$$(X_{t_1}^n,\dots ,X_{t_d}^n)\stackrel{d}{\to }(X_{t_1},\dots ,X_{t_d}).$$

proof

Let us define ${\pi }_{\tilde{t}}:C[0,\mathrm{\infty })\to {\mathbb{R}}^d$ as

$${\pi }_{\tilde{t}}(X(\omega )):=X_{\tilde{t}}(\omega )=(X_{t_1}(\omega ),\dots ,X_{t_d}(\omega )).$$

Note that ${\pi }_{\tilde{t}}$ is $\mathcal{B}(C[0,\mathrm{\infty }))/\mathcal{B}({\mathbb{R}}^d)$ measurable. Let $f\in C_b({\mathbb{R}}^d)$ be fixed. $f\circ {\pi }_{\tilde{t}}$ is bounded function from $C[0,\mathrm{\infty })$ to ${\mathbb{R}}^d$.

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim}{\int }_{{\mathrm{\Omega }}^n}f(X_{\tilde{t}}^n(\omega ))P_n(d\omega )& =& \underset{n\to \mathrm{\infty }}{lim}{\int }_{{\mathrm{\Omega }}^n}f({\pi }_{\tilde{t}}(X^n(\omega )))P_n(d\omega )\\ & =& \underset{n\to \mathrm{\infty }}{lim}{\int }_{{\mathrm{\Omega }}^n}(f\circ {\pi }_{\tilde{t}})(X^n(\omega )))P_n(d\omega )\\ & =& {\int }_{\mathrm{\Omega }}(f\circ {\pi }_{\tilde{t}})(X(\omega )))P(d\omega )\\ \text{(5)}& & =& {\int }_{\mathrm{\Omega }}f(X_{\tilde{t}}(\omega ))P(d\omega )\end{array}$$

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Theorem

• $(\mathrm{\Omega },\mathcal{F},P)$,
• $\{X^n{\}}_{n\in \mathbb{N}}$,
  • $X^n:{\mathrm{\Omega }}_n\to C[0,\mathrm{\infty })$,
  • ${\mathcal{F}}_n/\mathcal{B}(C[0,\mathrm{\infty }))$-measurable
  • tight sequence

$$\tilde{t}\in \mathcal{T},(X_{d_1}^n,\dots ,X_{t_d}^n)\stackrel{D}{\to }(X_{d_1},\dots ,X_{t_d}).$$

If

$$X^n\stackrel{d}{\to }X,$$

then for all $\tilde{t}:=(t_1,\dots ,t_d)\in \mathcal{T}$,

$$(X_{t_1}^n,\dots ,X_{t_d}^n)\stackrel{d}{\to }(X_{t_1},\dots ,X_{t_d}).$$

proof

$◻$

Theorem

• $(\mathrm{\Omega },\mathcal{F},P)$,
• $\{X^n{\}}_{n\in \mathbb{N}}$,
  • $X^n:{\mathrm{\Omega }}_n\to C[0,\mathrm{\infty })$,
  • ${\mathcal{F}}_n/\mathcal{B}(C[0,\mathrm{\infty }))$-measurable
  • tight sequence

$$\mathrm{\forall }\tilde{t}\in \mathcal{T},X_{\tilde{t}}^n\to X_{\tilde{t}}.$$

Let

$$\begin{array}{rcl}{P}^n& :=& P\circ (X^n{)}^{-1}:\mathcal{B}(C[0,\mathrm{\infty }))\to [0,1]\\ W_t(\omega )& :=& \omega (t)\end{array}.$$

Then there exists measure $P$ on $(C[0,\mathrm{\infty }),\mathcal{B}(C[0,\mathrm{\infty })))$ such that

$$\tilde{t}\in \mathcal{T},X_{\tilde{t}}^n\to W_{\tilde{t}}.$$

proof

By definition of tightness, every subsequence of tight sequence is also tight. Let $\overline{P}:=P\circ X^{-1}$. By Prohorov theorme, $\{P_n\}$ contains a weakly convergent subsequence. every subsequence of $\{P^n\}$ converges weakly to a probability measure $\overline{P}$.

$◻$

Reference