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$Y_{1}, Y_{2}, \ldots$ is a seq. of independent r.v.s, which meets the condition $\left|Y_{i}\right| \leq H$ for all $i$ for some positive $H$. If $\sum_{n=1}^{\infty} \operatorname{Var}\left(Y_{n}\right)=\infty .$ Define $S_{n}=Y_{1}+\cdots+Y_{n}$ for $n=1,2, \ldots$

How to prove that $$ \frac{S_{n}-\mathbb{E}\left[S_{n}\right]}{\sqrt{\operatorname{Var}\left(S_{n}\right)}} \stackrel{d}{\rightarrow} \mathcal{N}(0,1) $$ where $\mathcal{N}(0,1)$ denotes the standard normal distribution.

My idea: I want to use Lindeberg-Feller CLT to show this, but I am not sure how to check the condition.

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You can try $X_n = \frac{Y_n - \mathbb{E}\left[Y_{n}\right]}{Var(S_n)}$, then apply Lindeberg-Feller CLT

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