ThmDex – An index of mathematical definitions, results, and conjectures.
F11135
Formulation 0
Let $X_1, X_2, X_3, \ldots \in \mathsf{Random}(\mathbb{R})$ each be a D3161: Random real number such that
(i) \begin{equation} \forall \, n \in 1, 2, 3, \ldots : \mathbb{E} |X_n|^2 < \infty \end{equation}
(ii) $\lambda_1, \lambda_2, \lambda_3, \ldots \in \mathbb{R}$ are each a D993: Real number
(iii) \begin{equation} \lim_{n \to \infty} \frac{\mathsf{Var} X_n}{\lambda^2_n} = 0 \end{equation}
Then \begin{equation} \frac{X_n - \mathbb{E} X_n}{\lambda_n} \overset{p}{\longrightarrow} 0 \quad \text{ as } \quad n \to \infty \end{equation}