Tightness

Definiton 1

• $(S, \rho)$,
  • metric space
• $\Pi$,
  • a family of probability measures on $(S, \mathcal{B}(\mathbb{R}^{n}))$,

$\Pi$ is said to be relatively compact if

\[\forall \{P_{n}\}_{n \in \mathbb{N}} \subseteq \Pi, \ \exists \{P_{i_{n}}\}_{n \in \mathbb{N}} \subseteq \{P_{n}\}_{n}, \ \exists P \ \text{ s.t. } P_{i_{n}} \overset{w}{\rightarrow} P .\]

$\Pi$ is said to be tight if

\[\forall \epsilon > 0, \ \exists K \subseteq S \ \text{ s.t. } K \text{ is compact and } P(K) \ge 1 - \epsilon \ (\forall P \in \Pi) .\]

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Definition 2

• \((\Omega_{a}, \mathcal{F}_{a}, P_{a})\),
  • probabilty spaces
• \(\{X^{a}\}_{a \in \Lambda}\),
  • a family of $S$-valued r.v.s on $(\Omega_{a}, \mathcal{F}{a}, P{a})$

\(\{X^{a}\}_{a \in \Lambda}\) is said to be relatively compact if \(\{P(X^{a})^{-1}\}\) is relatively compact.

\(\{X^{a}\}_{a \in \Lambda}\) is said to be tight if \(\{P(X^{a})^{-1}\}\) is tight.

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Theorem 3 Prohorov Theorem

• $(S, \rho)$,
  • a compelte, separable metric space
• $\Pi$,
  • a family of probability measures on $S$,

Then the following statements are equivalent;

(1) $\Pi$ is tight

(2) $\Pi$ is realatively compact.

proof

$\Box$

Definiton 4

• $T > 0$,
• $\delta > 0$,
• $\omega \in C[0, \infty)$,

\[\begin{eqnarray} m^{T}(\omega, \delta) & := & \max_{\abs{s - t} \le \delta, 0 \le s, t \le T} \abs{ \omega(s) - \omega(t) } \nonumber \end{eqnarray}\]

is caleld the modulus of continuity on $[0, T]$.

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Proposition 5

• (1) $m^{T}$ is continuous in $\omega \in C[0, \infty)$ under the metric $\rho$ defined in here.
• (2) $m^{T}$ is non-decreasing in $\delta$
• (3) for each $\omega \in C[0, \infty)$, $\lim_{\delta \searrow 0} m^{T}(\omega, \delta) = 0$.

proof

$\Box$

Reference