A D5078: Random Euclidean real process $X : [0, \infty) \to \text{Random}(\mathbb{R}^N)$ is a standard Wiener process in dimension $N$ if and only if

(1)	\begin{equation} X(0) = \boldsymbol{0} \end{equation}
(2)	\begin{equation} X = (X_1, X_2, \, \ldots, \, X_N) \end{equation}
(3)	$X_1, X_2, \, \ldots, \, X_N$ are each a D3658: Standard real Wiener process
(4)	$X_1, X_2, \, \ldots, \, X_N$ is an D2713: Independent random collection