Large Deviations

This paper is based on Wald Lectures given at the annual meeting of the IMS in Minneapolis during August 2005. It is a survey of the theory of large deviations. 1. Large deviations for sums. The role of " large deviations " is best understood through an example. Suppose that X random variables, for instance, normally distributed with mean zero and variance 1. Then, E e θ(X 1 +···+X n) = E[e θX 1 ] n = e n(θ 2 /2). n is nearly zero, we have E e θ(X 1 +···+X n) = E e o(n) = e o(n). There is, of course, a very simple explanation for this. In computing expectations of random variables that can assume large values with small probabilities, contributions from such values cannot be ignored. After all, a product of something big and something small can still be big! In our case, assuming θ > 0, for any a > 0, E[e θS n ] ≥ e nθ a P S n n ≥ a = e nθ a e −(na 2 /2)+o(n) = e n(aθ −a 2 /2)+o(n). E[e θS n ] ≥ e n sup a>0 (θa−a 2 /2)+o(n) = e nθ 2 /2+o(n) , which is the correct answer. The simplest example for which one can calculate probabilities of large devia