ABSTRACT
This chapter contains the definition of the large deviations principle, as well as a few standard introductory results. The general theory of large deviations has a beautiful and powerful formulation due to Varadhan, called the “large deviations principle.” Varadhan's integral lemma and Sanov's theorem are standard tools of the large deviationist. The chapter first paraphrases the large deviations principle in the language of random variables. Note that there are no substantial assumptions made as to the range of these random variables, their distributions, or their dependencies. Chernoff’s Theorem is very useful in telling us how often certain rare events occur. Sanov’s Theorem tells us something amazing, and possibly more useful: it tells us how these events occur on the occasions when they do. The chapter also provides a first taste of “level 2 large deviations.” Its main purpose is to introduce the concept of large deviations of empirical measures.
