Don'sker's Theorem
Theorem
Let’s say $\left\{ \xi_i \right\}_{i \in \mathbb{N}}$ is a probability process defined in $(0,1)$. Suppose in the function space $C[0,1]$, the probability function $X_{n}$ is defined as follows: $$ X_{n}:= {{ 1 } \over { \sqrt{n} }} \sum_{i=1}^{\lfloor nt \rfloor} \xi_{i} + \left( nt - \lfloor nt \rfloor \right) {{ 1 } \over { \sqrt{n} }} \xi_{\lfloor nt \rfloor + 1} $$ $X_{n}$ converges in distribution to the Wiener Process $W$ when $n \to \infty$.
- $C[0,1]$ is a space of continuous functions with domain $[0,1]$ and codomain $\mathbb{R}$.
- $\lfloor \cdot \rfloor$ is known as the Floor Function, which denotes the value obtained by removing the decimal part in $\cdot$. In Korea, it is widely known as the Gauss function $[ \cdot ]$ in high schools.
Explanation
Donsker’s Theorem is also called Donsker’s invariance principle, functional central limit theorem, etc. Since the Wiener process feels like the normal distribution in the probability process, the fact that a probabilistic process, i.e., a probabilistic element, converges in distribution to the Wiener process is adequately termed as the Functional Central Limit Theorem.