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Definition D3658
Standard real Wiener process

Let $X_1, X_2, X_3, \ldots \in \text{Random} \{ -1, 1 \}$ each be a D5075: Random integer such that
 (i) $$\forall \, n \in \{ 1, 2, 3, \ldots \} : \mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = 1/2$$ (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 $$\forall \, t \in [0, \infty) : W_t \overset{d}{=} \lim_{N \to \infty} \frac{1}{\sqrt{N}} \sum_{n = 1}^{\lfloor N t \rfloor} X_n$$
Children
 ▶ Real Wiener process