### Large deviations

Here is a nice theorem about Gaussian tails. Suppose $Y_n = \sum_{i=1}^n X_{i,n}$ where $X_{i,n}$ are mutually independent random variables with mean $\mathrm{E}[X_{i,n}] = 0$ and variance $\mathrm{Var}[X_{i,n}] = \sigma_{i,n}^2$. Denote $\mathrm{Var}[Y_n] = \sigma_n^2 = \sum_{i=1}^n \sigma_{i,n}^2$. We consider the probability $\mathrm{P}[Y_n \geq a_n \sigma_n]$ as $\lim_{n\rightarrow \infty} \lambda_n \equiv \frac{a_n}{\sigma_n} = 0$, that is, $\sigma_n$ goes to infinity faster than $a_n$. Assume further that:

$\mathrm{E}[\exp(\lambda_n X_{i,n})] \leq \exp((1+o(1)) \lambda_n^2 \sigma_{i,n}^2 / 2)$

A simple sufficient condition for this last equation to hold is that the $X_{i,n}$ be bounded, i.e. that there be a constant $K$ such that $|X_{i,n}|\leq K$ for all $i$ and $n$. Then the following theorem holds (the proof is elementary and can be found in this book):

Theorem: $\mathrm{P}[Y_n \geq a_n \sigma_n] \leq \exp(- (1+o(1))a_n^2 / 2)$

This is exactly the tail behavior of a Gaussian random variable. Hence, this theorem shows that under somewhat general conditions, tails of sums of independent random variables asymptotically behave like Gaussian tails.