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
\[\forall f \in C_{b}(S), \ \lim_{n \rightarrow \infty} \int_{S} f(s) \ P_{n}(ds) = \int_{S} f(s) \ P(ds) .\]We write $P_{n} \overset{w}{\rightarrow} P$.
Proposition 1
- $(S, \rho)$,
- $\mathcal{O}$,
- all open sets in $S$,
(1)
If $P_{n} \overset{w}{\rightarrow} P$, $P_{n} \overset{w}{\rightarrow} P^{\prime}$, then $P = P^{\prime}$.
(2) Let $F$ be closed set.
\[\begin{eqnarray} f(x) & := & (1 - \rho(x, F)/\epsilon)^{+} \end{eqnarray}\](2-1) $f$ is bounded.
(2-2) $f$ is uniformly continuous
(2-3)
\[\begin{eqnarray} I_{F}(x) \le f(x) \le I_{F^{\epsilon}}(x) . \end{eqnarray}\]proof
proof of (1)
\[\begin{eqnarray} \forall O \in \mathcal{O}, \ P(O) & = & \int_{S} 1_{O}(x) \ P(dx) \nonumber \\ & = & \lim_{n \rightarrow \infty} \int_{S} 1_{O}(x) \ P_{n}(dx) \nonumber \\ & = & \int_{S} 1_{O}(x) \ P^{\prime}(dx) \nonumber \\ & = & P^{\prime}(O) \nonumber \end{eqnarray}\]proof of (2)
Since $0 \le f_{\epsilon} \le 1$, (2-1) holds.
As to (2-2),
\[\begin{eqnarray} \abs{ f_{\epsilon}(x) - f_{\epsilon}(x) } \le \end{eqnarray}\]Definition 3
- $(S, \rho)$,
- metric space
- \(\{(\Omega_{n}, \mathcal{F}_{n}, P_{n})\}_{n \in \mathbb{N}}\),
- a sequence of probability spaces
- $X_{n}: \Omega \rightarrow S$,
- random variable over \((\Omega_{n}, \mathcal{F}_{n}, P_{n})\),
- $(\Omega, \mathcal{F}, P)$,
- probability space
- random variable on \((\Omega_{n}, \mathcal{F}_{n}, P_{n})\),
- $X: \Omega \rightarrow S$,
- random variable over $(\Omega, \mathcal{F}, P)$,
\(\{X_{n}\}\) is said to converge to $X$ in distribution if
\[\begin{eqnarray} & & P_{n}X_{n}^{-1} \overset{w}{\rightarrow} PX^{-1} \nonumber \\ & \Leftrightarrow & \forall f \in C_{b}(S), \ \lim_{n \rightarrow \infty} \int_{S} f(s) \ (P_{n}X_{n}^{-1})(ds) = \int_{S} f(s) \ (PX^{-1})(ds) . \end{eqnarray} .\]We write $X_{n} \overset{d}{\rightarrow} X$.
Proposition 4
(1)
$X_{n} \overset{d}{\rightarrow} X$ if and only if
\[\forall f \in C_{b}(S), \ \lim_{n \rightarrow \infty} \int_{S_{n}} f(X_{n}(\omega)) \ P_{n}(d \omega) = \int_{S} f(X(\omega)) \ P(d \omega) .\]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.
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) $P_{n} \overset{w}{\rightarrow} P$,
- (ii) $P_{n} f \rightarrow Pf$ for all bounded, uniformly continuous $f$,
- (iii) $\lim\sup_{n} P_{n}F \le PF$ for all $F \in \mathcal{F}$
- (iv) $\lim\inf_{n} P_{n}G \ge PG$ for all $G \in \mathcal{O}$
- (v) $P_{n}A \rightarrow PA$ for all $P$ continuity sets $A$
proof
(i) $\Rightarrow$ (ii)
(ii) $\Rightarrow$ (iii)
(iii) $\Leftrightarrow$ (iv)
(iii) $\Rightarrow$ (v)
(v) $\Rightarrow$ (i)
Theorem 7
- $(\Omega, \mathcal{F}, P)$,
- $X:\Omega \rightarrow C[0, \infty)$,
- $\mathcal{F}/\mathcal{B}(C[0, \infty))$-measurable
- \((\Omega_{n}, \mathcal{F}_{n}, P_{n})\),
- probability spaces
- $X^{n}:\Omega_{n} \rightarrow C[0, \infty)$,
- $\mathcal{F}_{n}/\mathcal{B}(C[0, \infty))$-measurable
If
\[X^{n} \overset{d}{\rightarrow} X,\]then for all $\tilde{t} := (t_{1}, \ldots, t_{d}) \in \mathcal{T}$,
\[(X_{t_{1}}^{n}, \ldots, X_{t_{d}}^{n}) \overset{d}{\rightarrow} (X_{t_{1}}, \ldots, X_{t_{d}}) .\]proof
Let us define $\pi_{\tilde{t}}: C[0, \infty) \rightarrow \mathbb{R}^{d}$ as
\[\pi_{\tilde{t}}(X(\omega)) := X_{\tilde{t}}(\omega) = (X_{t_{1}}(\omega), \ldots, X_{t_{d}}(\omega)) .\]Note that $\pi_{\tilde{t}}$ is $\mathcal{B}(C[0, \infty))/\mathcal{B}(\mathbb{R}^{d})$ measurable. Let $f \in C_{b}(\mathbb{R}^{d})$ be fixed. $f \circ \pi_{\tilde{t}}$ is bounded function from $C[0, \infty)$ to $\mathbb{R}^{d}$.
\[\begin{eqnarray} \lim_{n \rightarrow \infty} \int_{\Omega^{n}} f(X_{\tilde{t}}^{n}(\omega)) \ P_{n}(d \omega) & = & \lim_{n \rightarrow \infty} \int_{\Omega^{n}} f(\pi_{\tilde{t}}(X^{n}(\omega))) \ P_{n}(d \omega) \nonumber \\ & = & \lim_{n \rightarrow \infty} \int_{\Omega^{n}} (f \circ \pi_{\tilde{t}})(X^{n}(\omega))) \ P_{n}(d \omega) \nonumber \\ & = & \int_{\Omega} (f \circ \pi_{\tilde{t}})(X(\omega))) \ P(d \omega) \nonumber \\ & = & \int_{\Omega} f(X_{\tilde{t}}(\omega)) \ P(d \omega) \end{eqnarray}\]Theorem
- $(\Omega, \mathcal{F}, P)$,
- \(\{X^{n}\}_{n \in \mathbb{N}}\),
- $X^{n}:\Omega_{n} \rightarrow C[0, \infty)$,
- \(\mathcal{F}_{n}/\mathcal{B}(C[0, \infty))\)-measurable
- tight sequence
If
\[X^{n} \overset{d}{\rightarrow} X,\]then for all $\tilde{t} := (t_{1}, \ldots, t_{d}) \in \mathcal{T}$,
\[(X_{t_{1}}^{n}, \ldots, X_{t_{d}}^{n}) \overset{d}{\rightarrow} (X_{t_{1}}, \ldots, X_{t_{d}}) .\]proof
Theorem
- $(\Omega, \mathcal{F}, P)$,
- \(\{X^{n}\}_{n \in \mathbb{N}}\),
- $X^{n}:\Omega_{n} \rightarrow C[0, \infty)$,
- $\mathcal{F}_{n}/\mathcal{B}(C[0, \infty))$-measurable
- tight sequence
Let
\[\begin{eqnarray} P^{n} & := & P \circ (X^{n})^{-1}: \mathcal{B}(C[0, \infty)) \rightarrow [0, 1] \nonumber \\ W_{t}(\omega) & := & \omega(t) \nonumber \end{eqnarray} .\]Then there exists measure $P$ on \((C[0, \infty), \mathcal{B}(C[0, \infty)))\) such that
\[\tilde{t} \in \mathcal{T}, \ X_{\tilde{t}}^{n} \rightarrow W_{\tilde{t}} .\]proof
By definition of tightness, every subsequence of tight sequence is also tight. Let $\bar{P} := P \circ X^{-1}$. By Prohorov theorme, \(\{P_{n}\}\) contains a weakly convergent subsequence. every subsequence of \(\{P^{n}\}\) converges weakly to a probability measure $\bar{P}$.