Large deviation theory and applications

Large deviation theory deals with the decay of the probability of increasingly unlikely events. It is one of the key techniques of modern probability, a role which is emphasised by the recent award of the Abel prize to S.R.S. Varadhan, one of the pioneers of the subject. The subject is intimately related to combinatorial theory and the calculus of variations. Applications of large deviation theory arise, for example, in statistical mechanics, information theory and insurance. 1 Cramér’s theorem and the moderate deviation principle We start by looking at an example embedded in the most classical results of probability theory. Suppose that X and X1, X2, . . . are independent, identically distributed random variables with mean μ and variance σ < ∞. We denote the partial sum by Sn := n ∑