Let $X_1,X_2,\dots\,$ i.i.d random variables with mean zero and variance $1$. Let $S_n=\sum_{i=1}^n X_i\,,n\in \mathbb N.$ Compute the weak limes $\lim_{n\to\infty} \frac1n \sum_{i=1}^n \frac{S_i}{\sqrt n}$
Surely we will have to use the CLT. First I tried to simplify the expression, but I am not sure how to continue here. $$\lim_{n\to\infty} \frac1n \sum_{i=1}^n \frac{S_i}{\sqrt n}=\dots=\lim_{ n\to\infty}\frac{1}{\sqrt n} \frac{nX_1+(n-1)X_2+\dots+X_n}{n}$$
Edit(2)
According to the comments, we have to verify Lindberg's condition (https://en.wikipedia.org/wiki/Lindeberg%27s_condition)
Lindberg's condition: $$\lim_{n\to\infty} \frac{1}{s_n^2} \sum_{k=1}^n E[(X_k - \mu_k)^2 \mathbb 1_{\{\mid X_k - \mu_k \mid > \epsilon s_n \}}=0,\quad \text{for all $\epsilon >0$}$$
Here:
$E(S_i) {\overset{\text{$X_i$ i.i.d}}{=}}0$ , $Var(S_i) {\overset{\text{$X_i$ i.i.d}}{=}} \sum Var( X_i) {\overset{\text{$X_i$ i.i.d}}{=}} i$ for all $i=1,2,\dots$ Furthermore $s_n^2= \sum_{i=1}^n \sigma_i^2 =Var(S_1)+Var(S_2)+\dots + Var(S_n)=1+2+\dots +n=\frac{n(n+1)}{2}$.
Plugging in: $$\lim_{n\to\infty}\frac{2}{n^2+n}\sum_{k=1}^n E(S_k)^2 1_{\{\mid S_k \mid > \epsilon {\frac{\sqrt {n^2+n}}{\sqrt 2}}\}}$$ Intuitively this does not seem correct to me. Furthermore I am not sure how to simplify this expression.
Some help is welcome and obviously needed!
Lindeberg's condition
in your favorite search engine and see what pops up... $\endgroup$