ThmDex – An index of mathematical definitions, results, and conjectures.
Weak law of large numbers for variance with weighted decay

Let $X_1, X_2, X_3, \ldots \in \mathsf{Random}(\mathbb{R})$ each be a D3161: Random real number such that
 (i) $$\forall \, n \in 1, 2, 3, \ldots : \mathbb{E} |X_n|^2 < \infty$$ (ii) $\lambda_1, \lambda_2, \lambda_3, \ldots \in \mathbb{R}$ are each a D993: Real number (iii) $$\lim_{n \to \infty} \frac{\mathsf{Var} X_n}{\lambda^2_n} = 0$$
Then $$\frac{X_n - \mathbb{E} X_n}{\lambda_n} \overset{p}{\longrightarrow} 0 \quad \text{ as } \quad n \to \infty$$
Proofs
Proof 0
Let $X_1, X_2, X_3, \ldots \in \mathsf{Random}(\mathbb{R})$ each be a D3161: Random real number such that
 (i) $$\forall \, n \in 1, 2, 3, \ldots : \mathbb{E} |X_n|^2 < \infty$$ (ii) $\lambda_1, \lambda_2, \lambda_3, \ldots \in \mathbb{R}$ are each a D993: Real number (iii) $$\lim_{n \to \infty} \frac{\mathsf{Var} X_n}{\lambda^2_n} = 0$$
By hypothesis (iii), we have $$\mathbb{E} \left| \frac{X_n - \mathbb{E} X_n}{\lambda_n} \right| = \frac{1}{\lambda^2_n} \mathbb{E} |X_n - \mathbb{E} X_n|^2 = \frac{1}{\lambda^2_n} \mathsf{Var} X_n \longrightarrow 0$$ as $n \to \infty$. This shows that $(X_n - \mathbb{E} X_n) / \lambda_n \overset{L^2}{\longrightarrow} 0$ and the result is now a consequence of R3250: P-convergence implies convergence in probability. $\square$