I want to find the limit of expressions such as: $\lim_{n \to \infty} P(\frac{|S_{n}|}{n} \le \epsilon) $. I do not know how to proceed.
Using the central limit theorem, I can rewrite the expression: $$\lim_{n \to \infty} P(|S_{n}|\le n \cdot \epsilon) = \lim_{n \to \infty} P(- n \epsilon \le |S_{n}| \le n \epsilon) = \int_{-\infty}^{\infty}\frac{1}{2\pi}e^{-x^{2}/2}=1$$
Is that correct?
If so, how do I proceed in the case $\lim_{n \to \infty} P(\frac{|S_{n}|}{n} =0) $?
Your support is very much appreciated! Thank you!
Best, Jolle