Standard gaussian random real number

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 D3161: Random real number $Z \in \text{Random}(\mathbb{R})$ is a standard gaussian random real number if and only if $$Z \overset{d}{=} \lim_{N \to \infty} \sum_{n = 1}^N \frac{X_n}{\sqrt{N}}$$

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 D3161: Random real number $Z \in \text{Random}(\mathbb{R})$ is a standard gaussian random real number if and only if $$Z \overset{d}{=} \lim_{N \to \infty} \left( \frac{X_1}{\sqrt{N}} + \frac{X_2}{\sqrt{N}} + \cdots + \frac{X_N}{\sqrt{N}} \right)$$

Let $X_1, X_2, X_3, \ldots \in \text{Rademacher}(1 / 2)$ each be a D5287: Standard rademacher random integer such that
 (i) $X_1, X_2, X_3, \ldots$ is an D2713: Independent random collection
A D3161: Random real number $Z \in \text{Random}(\mathbb{R})$ is a standard gaussian random real number if and only if $$Z \overset{d}{=} \lim_{N \to \infty} \sum_{n = 1}^N \frac{X_n}{\sqrt{N}}$$
Also known as
Standard normal random real number
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