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2014 | OriginalPaper | Chapter
23. Large Deviations
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Abstract
The central limit theorem measures the typical fluctuations of sums of (square integrable) random variables. On the other hand, laws of large numbers only show that atypically large deviations have vanishing probabilities (which is just another way of saying that they are atypical).
The aim of this chapter is to give more precise quantitative statements about the small probabilities of atypically large fluctuations (large deviations). We present Cramér’s theorem on large deviations for sums of real random variables as well as Sanov’s theorem on large deviations for empirical distributions in discrete models. Finally, we use Varadhan’s lemma to provide a connection to statistical physics and, in particular, the Weiss ferromagnet.