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I need to calclulate $$\lim_{n\to \infty} \frac{1}{n}\sum_{i=1}^n \frac{S_i }{\sqrt{n}}$$ where $$ S_i=\sum_{i=1}^n X_i$$ for i.i.d $X_1,X_2,...$ with Expectation 0 and variance 1. I know from the CLT that $\sum_{i=1}^n \frac{X_i }{\sqrt{n}}$ converges to the Standard-Normal distribution. Note that I need to calculate the weak limit. What does that even mean?

Solved by JGWang


marked as duplicate by Davide Giraudo, Did, Shuhao Cao, man and laptop, José Carlos Santos Jan 23 '18 at 23:35

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