ThmDex – An index of mathematical definitions, results, and conjectures.
I.I.D. strong law of large numbers for random real numbers of finite fourth moments
Formulation 0
Let $X_1, X_2, X_3, \ldots \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number such that
(i) $X_1, X_2, X_3, \ldots$ is an D3358: I.I.D. random collection
(ii) \begin{equation} \mathbb{E} |X_1|^4 < \infty \end{equation}
(iii) \begin{equation} \mu : = \mathbb{E} X_1 \end{equation}
Then \begin{equation} \sum_{n = 1}^N \frac{X_n - \mu}{N} \overset{a.s.}{\longrightarrow} 0 \quad \text{ as } \quad N \to \infty \end{equation}
Formulation 1
Let $X_1, X_2, X_3, \ldots \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number such that
(i) $X_1, X_2, X_3, \ldots$ is an D3358: I.I.D. random collection
(ii) \begin{equation} \mathbb{E} |X_1|^4 < \infty \end{equation}
(iii) \begin{equation} \mu : = \mathbb{E} X_1 \end{equation}
Then \begin{equation} \mathbb{P} \left( \lim_{N \to \infty} \sum_{n = 1}^N \frac{X_n - \mu}{N} = 0 \right) = 1 \end{equation}