# CLT Application: converge in distribution to N(0,1)

$$Y_{1}, Y_{2}, \ldots$$ 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.

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