Sanov's theorem of large deviations.

I don't have to prove anything, right? If they want a proof, they'll look it up in a book later.

Assume the students already know about the central limit theorem. Explain how the two theorems talk about limits in different direction: let $ S_n $ be the sum of $ n $ independent variables of identical distributions (real valued, with zero mean and finite variance), the central limit theorem gives a limit of the unscaled probability $ P(S_n/\sqrt{n} < c) $, this limit is strictly between 0 and 1; whereas large deviation theorems give the rate of decrease of a probability like $ P(S_n/n < c) $.