Let $(Z_{n,i} ; 1 \leq i \leq n)_{n \geq 1}$ be a triangular array of i.i.d. random variables with $\operatorname{Var}(Z_{n,i}) = \sigma_n^2 \to \sigma^2$ as $n \to \infty$. Suppose the following conditions hold: 1. For all $\delta > 0$: $n \cdot \mathbb{P}(|Z_{n,1}| > \delta \sqrt{n}) \to 0$ as $n \to \infty$. 2. $\operatorname{Var}(Z_{n,1} \cdot \mathbf{1}_{\{|Z_{n,1}| \leq \sqrt{n}\}}) \to \sigma^2$ as $n \to \infty$. 3. $\sqrt{n}\, \mathbb{E}[Z_{n,1} \cdot \mathbf{1}_{\{|Z_{n,1}| > \sqrt{n}\}}] \to 0$ as $n \to \infty$. Then \begin{align*} \sqrt{n}\left(\frac{1}{n} \sum_{i=1}^n Z_{n,i} - \mathbb{E}_n[Z_{n,i}]\right) \xrightarrow{d} N(0, \sigma^2). \end{align*}