Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of independent random variables, each with standard Gaussian distribution. For a given $K>0$, prove that:
$$\lim_{n\to\infty} \frac{1}{n}\log{P\left(\left |\frac{X_1+\,...\,+X_n}{n}\right| \geq K\right)}=-\frac{K^2}{2} $$
I don't know how I can deal with this logarithm. I tried to rewrite the equation as:
$$\lim_{n\to\infty} \left(P\left(\left |\frac{X_1+\,...\,+X_n}{n}\right| \geq K\right)\right)^{\frac{1}{n}}=e^{-\frac{K^2}{2}}$$
so that now I could get rid of the logarithm and get a nice looking $e^{-\frac{K^2}{2}}$. But I actually don't know what to do with that. Any ideas?
Thanks!