Large deviations and applications

Large deviations is concerned with the study of rare events and of small probabilities. Let Xi,  ≤ i ≤ n, be independent identically distributed (i.i.d.) real random variables with expectationm, and Xn = (X + . . . +Xn)/n their empirical mean. e law of large numbers shows that, for any Borel set A ⊂ R not containing m in its closure, P(Xn ∈ A) →  as n → ∞, but does not tell us how fast the probability vanishes. Large deviations theory gives us the rate of decay, which is exponential in n. Cramer’s theorem states that,