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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

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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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