Let $X_1, X_2, X_3, \ldots \in \text{Random} \{ -1, 1 \}$ each be a
D5075: Random integer such that
| (i) |
\begin{equation}
\forall \, n \in \{ 1, 2, 3, \ldots \} :
\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = 1/2
\end{equation}
|
| (ii) |
$X_1, X_2, X_3, \ldots$ is an D2713: Independent random collection
|
A
D5076: Random real process $W : [0, \infty) \to \text{Random}(\mathbb{R})$ is a
standard Wiener process if and only if
\begin{equation}
\forall \, t \in [0, \infty) :
W_t
\overset{d}{=} \lim_{N \to \infty} \frac{1}{\sqrt{N}} \sum_{n = 1}^{\lfloor N t \rfloor} X_n
\end{equation}
