# Central Limit Theorem for not identical distributed but independent centered random variables with variance one.

so let's assume we have independent random variables $$X_1,X_2, X_3, \ldots$$ with $$\mathbb{E}[X_k]=0 \mbox{ and } \mathbb{Var}[X_k]=\sigma_k^2=1 \quad \forall k\in\mathbb{N}.$$

We define $$s_n^2:= \sum_{k=1}^n \mathbb{E}[(X_k-\mathbb{E}[X_k])^2]=\sum_{k=1}^n\sigma_k^2=n.$$

Now we check Lindenberg's condition: So for $$\varepsilon>0$$ we must check $$\lim\limits_{n\rightarrow \infty} \frac{1}{s_n^2}\sum_{k=1}^n \mathbb{E}[(X_k-\mathbb{E}[X_k])^2 \cdot \mathbf{1}_{|X_k| > \varepsilon s_n^2} ] =\lim\limits_{n\rightarrow \infty} \frac{1}{n} \sum_{k=1}^n \mathbb{E}[X_k^2 \cdot \mathbf{1}_{|X_k| > \varepsilon n} ].$$

We know, $$\mathbb{E}[X_k^2 ]=1,$$ so $$\mathbb{E}[X_k^2 \cdot \mathbf{1}_{|X_k| > \varepsilon n} ]$$ converges quicker than $$\frac{1}{n},$$ whch means we find an $$N\in\mathbb{N}$$ such that for every $$n\geq N$$ we have
$$\mathbb{E}[X_k^2 \cdot \mathbf{1}_{|X_k| > \varepsilon n} ]< \frac{1}{n},$$ which means for every $$\varepsilon>0$$ we have $$\lim\limits_{n\rightarrow \infty} \frac{1}{n} \sum_{k=1}^n \mathbb{E}[X_k^2 \cdot \mathbf{1}_{|X_k| > \varepsilon n} ] \leq \lim\limits_{n\rightarrow \infty} \frac{N}{n} + \frac{1}{n} \sum_{k=N+1}^n \mathbb{E}[X_k^2 \cdot \mathbf{1}_{|X_k| > \varepsilon n} ]< \lim\limits_{n\rightarrow \infty} \frac{N}{n} + + \frac{1}{n^2 }=0,$$ so we know that the Central Limit Theorem holds.

Is that right?