Large deviations theory

Large deviations theory is a branch of probability theory that deals with the study of rare events—specifically, events that deviate significantly from expected behavior. It provides a mathematical framework for quantifying the probabilities of these rare deviations from the average or typical outcome of a stochastic process. The fundamental ideas in large deviations theory include: 1. **Rate Functions**: These are functions that describe the exponential decay rate of the probabilities of rare events.

Cramér's theorem is a fundamental result in the field of large deviations theory, which examines the asymptotic behavior of the probabilities of rare events. Specifically, Cramér's theorem provides a way to quantify the likelihood of deviations of a sum of independent random variables from its expected value. The theorem states that if we have a sequence of independent and identically distributed (i.i.d.