ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Subset
Power set
Hyperpower set sequence
Hyperpower set
Hypersubset
Subset algebra
Subset structure
Topological space
Continuous map
Continuous function
Right-continuous function
Almost surely right-continuous random function
Cadlag random function
Lévy process
Definition D3658
Standard real Wiener process
Formulation 8
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 real 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}
Children
D3657: Real Wiener process