Schilder's theorem

Schilder's theorem is a fundamental result in probability theory, particularly in the area of large deviations. It provides an asymptotic estimate for the probabilities of large deviations for sequences of random variables. Specifically, it deals with the behavior of the empirical measures of random walks. More formally, Schilder's theorem states that for a sequence of independent and identically distributed random variables, the probability that the empirical measure deviates significantly from its expected value decays exponentially as the number of samples increases.