Large deviations
Srinivasa R. S. Varadhan · The Annals of Probability · 2008
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. Large deviations for sums.The role of "large deviations" is best understood through an example.Suppose that X 1 , X 2 , . . ., X n , . . . is a sequence of i.i.d.random variables, for instance, normally distributed with mean zero and variance 1.Then,Since, by the law of large numbers, S n n is nearly zero, we haveThere 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,Since a > 0 is arbitrary,which is the correct answer.The simplest example for which one can calculate probabilities of large devia-