Theorem 7.2

See modulus of continuity.

proof

$\Box$

Theorem 7.3

• \(\{P_{n}\}\),
  • a sequence of probablity measuares on \((C[0, \infty), \mathcal{B}(C[0, \infty))\),
• (1) \(\{P_{n}\}\) is tight,
• (2) (2-i) and (2-ii) holds

(2-i)

\[\begin{eqnarray} \forall > \eta, \ \exists a \in \mathbb{R}, \ n_{0} \in \mathbb{N} \text{ s.t. } \left( \forall n \ge n_{0}, \ P_{n}(x \in S \mid \abs{x(0)} \ge a) \le \eta \right) . \end{eqnarray}\]

(2-ii)

\[\forall \eta > 0, \ \forall \epsilon > 0, \ \exists \delta \in (0, 1), \ \exists n_{0} \in \mathbb{N} \text{ s.t. } \left( \forall n \ge n_{0}, \ P_{n} ( \omega \in C[0, \infty) \mid m^{T}(\omega, \delta) \ge \epsilon ) \le \eta \right) .\]

proof

(1) $\Rightarrow$ (2)

Let $T \in \mathbb{N}$ be fixed. Let $\eta > 0$ and $\epsilon > 0$ be fixed. By defintion of tightness, there exists compact set $K$ such that

\[\begin{eqnarray} \forall n \in \mathbb{N}, \ P_{n}(K) > 1 - \eta . \end{eqnarray}\]

Since $K$ is compact, by Theorem 7.2,

\[\begin{eqnarray} & & \sup_{\omega \in K} \abs{ \omega(0) } < \infty \nonumber \\ \forall T > 0, & & \lim_{\delta \rightarrow 0} \sup_{\omega \in K} m^{T}(\omega, \delta) = 0 . \nonumber \end{eqnarray}\]

Hence there exists $a \in \mathbb{R}$ such that

\[K \subseteq \{ \omega \in C[0, \infty) \mid \abs{\omega(0)} \le a \} .\]

Moreover, since for some $\delta > 0$

\[\begin{eqnarray} \sup_{\omega \in K} m^{T}(\omega, \delta) < \epsilon, \end{eqnarray}\]

there exists $\delta > 0$ such that

\[K \subseteq \{ \omega \in C[0, \infty) \mid m^{T}(\omega, \delta) < \epsilon \} .\]

Thus,

\[\begin{eqnarray} \eta & \ge & P(K^{c}) \nonumber \\ & \ge & P \left( \omega \in C[0, \infty) \mid m^{T}(\omega, \delta) \ge \epsilon \right) . \nonumber \end{eqnarray}\]

Similary,

\[\begin{eqnarray} \eta & \ge & P(K^{c}) \nonumber \\ & \ge & P \left( \omega \in C[0, \infty) \mid \abs{\omega(0)} \ge \epsilon \right) . \nonumber \end{eqnarray}\]

(1) $\Leftarrow$ (2)

$\Box$