Donsker Invariance Principle

Definitoin 1

• $(\Omega, \mathcal{F}, P)$,
• $\sigma^{2} \in (0, \infty)$,
• \(\{\xi_{j}\}_{j \in \mathbb{N}\),
  • I.I.D. sequence
  • mean is 0
  • variance is $\sigam$,

\[\begin{eqnarray} S_{k} & := & \begin{cases} 0 & k = 0 \\ \sum_{j=1}^{k} \xi_{j} & \text{otherwise} \end{cases} \nonumber \\ Y_{t}^{n} & := & S_{\lfloor t \rfloor} + (t - \lfloor t \rfloor) \xi_{\lfloor t \rfloor + 1} \ t \ge 0 \label{donsker_invariance_principle_def_linear_interpolation} \\ X_{t}^{n} & := & \frac{1}{\sigma \sqrt{n}} Y_{nt} \label{donsker_invariance_principle_def_scale} \\ P_{n}: \mathcal{B}(C[0, \infty)) \rightarrow [0, 1], \ P_{n} & := & P(X^{n})^{-1} \label{donsker_invariance_principle_def_induced_measure} . \end{eqnarray}\]

■

Theorem 1

proof

$\Box$

Theorem The Invariance Principle of Donsker

• $(\Omega, \mathcal{F}, P)$,
• $\sigma^{2} \in (0, \infty)$,
• \(\{\xi_{j}\}_{j \in \mathbb{N}\),
  • I.I.D. sequence
  • mean is 0
  • variance is $\sigam$,

Then \(\{P_{n}\}_{n \in \mathbb{N}}\) converges weakly to a measure $P_{*}$ under which the coordinate mapping process $W_{t}(\omega) := \omega(t)$ is a standard one-dimensional Browninan motion.

proof

$\Box$

Reference

• Donsker’s theorem - Wikipedia