Large deviations theory

Summary

Large deviations theory studies the exponential decay of the probabilities of rare events. Its central notion is the large deviation principle, with a rate function $I$. A family of probability measures $\mu_n$ satisfies it when $\mu_n(A)$ behaves like $\exp(-n \inf_{x \in A} I(x))$. Basic results are Cramér's theorem for sample means, Sanov's theorem for empirical measures and Varadhan's lemma for exponential integrals. The theory is used in statistical mechanics, information theory, queueing and risk analysis.