ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5404 on D3358: I.I.D. random collection
I.I.D. real central limit theorem
Formulation 0
Let $X_1, X_2, X_3, \dots \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number such that
(i) $X_1, X_2, X_3, \dots$ is an D3358: I.I.D. random collection
(ii) \begin{equation} \mathbb{E} |X_1|^2 \in (0, \infty) \end{equation}
(iii) \begin{equation} \mu : = \mathbb{E} X_1 \end{equation}
(iv) \begin{equation} \sigma^2 : = \text{Var} X_1 \end{equation}
Let $f : \{ 1, 2, 3, \ldots \} \to \{1, 2, 3, \ldots \}$ be a D5406: Positive integer function such that
(i) \begin{equation} \lim_{N \to \infty} f(N) = \infty \end{equation}
Then \begin{equation} \lim_{N \to \infty} \sum_{n = 1}^{f(N)} \frac{X_n - \mu}{\sqrt{\sigma^2 f(N)}} \overset{d}{=} \text{Gaussian}(0, 1) \end{equation}
Formulation 1
Let $X_1, X_2, X_3, \dots \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number such that
(i) $X_1, X_2, X_3, \dots$ is an D3358: I.I.D. random collection
(ii) \begin{equation} \mathbb{E} |X_1|^2 \in (0, \infty) \end{equation}
(iii) \begin{equation} \mu : = \mathbb{E} X_1 \end{equation}
(iv) \begin{equation} \sigma^2 : = \text{Var} X_1 \end{equation}
Let $f : \{ 1, 2, 3, \ldots \} \to \{1, 2, 3, \ldots \}$ be a D5406: Positive integer function such that
(i) \begin{equation} \lim_{N \to \infty} f(N) = \infty \end{equation}
Then \begin{equation} \sum_{n = 1}^{f(N)} \frac{X_n - \mu}{\sqrt{\sigma^2 f(N)}} \overset{d}{\longrightarrow} \text{Gaussian}(0, 1) \quad \text{ as } \quad N \to \infty \end{equation}
Proofs
Proof 0
Let $X_1, X_2, X_3, \dots \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number such that
(i) $X_1, X_2, X_3, \dots$ is an D3358: I.I.D. random collection
(ii) \begin{equation} \mathbb{E} |X_1|^2 \in (0, \infty) \end{equation}
(iii) \begin{equation} \mu : = \mathbb{E} X_1 \end{equation}
(iv) \begin{equation} \sigma^2 : = \text{Var} X_1 \end{equation}
Let $f : \{ 1, 2, 3, \ldots \} \to \{1, 2, 3, \ldots \}$ be a D5406: Positive integer function such that
(i) \begin{equation} \lim_{N \to \infty} f(N) = \infty \end{equation}
Result R3681: Mean-deviance standardisation of a random real number shows that the random real number \begin{equation} \frac{X_n - \mu}{\sigma} \end{equation} has expectation $0$ and variance $1$. Hence, the result follows from R5405: Standard I.I.D. real central limit theorem. $\square$