Foundations of Modern Probability pp 537-560 | Cite as

# Large Deviations

Chapter

## Abstract

Legendre-Fenchel transform; Cramér’s and Schilder’s theorems; large-deviation principle and rate function; functional form of the LDP; continuous mapping and extension; perturbation of dynamical systems; empirical processes and entropy; Strassen’s law of the iterated logarithm

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## References

- Large deviation theory originated with certain refinements of the central limit theorem obtained by many authors, beginning with Khinchin (1929). Here the object of study is the ratio of tail probabilities (math), where ζ is
*N*(0,1) and(math) for some i.i.d. random variables & with mean 0 and variance 1, so that r_{n}(*x*) → 1 for fixed*x.*A precise asymptotic expansion was obtained by Cramer (1938), in the case when*x*varies with*n*at a rate x = 0(n^{1/2}). (See Petrov (1995), Theorem 5.23, for details.)Google Scholar - In the same historic paper, Cramer (1938) obtained the first true large deviation result, in the form of our Theorem 27.3, though under some technical assumptions that were later removed by Chernoff (1952) and Bahadur (1971). Varadhan (1966) extended the result to higher dimensions and rephrased it in the form of a general large deviation principle. At about the same time, Schilder (1966) proved his large deviation result for Brownian motion, using the present change-of-measure approach. Similar methods were used by Freidlin and Wentzell (1970, 1998) to study random perturbations of dynamical systems.Google Scholar
- Even earlier, Sanov (1957) had obtained his large deviation result for empirical distributions of i.i.d. random variables. The relative entropy
*H(v\µ*) appearing in the limit had already been introduced in statistics by Kullback and Leibler (1951). Its crucial link to the Legendre-Fenchel transform Λ, long anticipated by physicists, was formalized by Donsker and Varadhan (1975–83). The latter authors also developed some profound and far-reaching extensions of Sanov’s theorem, in a long series of formidable papers. Ellis (1985) gives a detailed exposition of those results, along with a discussion of their physical significance.Google Scholar - Much of the formalization of underlying principles and techniques was developed at a later stage. Thus, an abstract version of the projective limit approach was introduced by Dawson and Gärtner (1987). BrycGoogle Scholar
- (1990) supplemented Varadhan’s (1966) functional version of the LDP with a reverse proposition. Similarly, Ioffe (1991) appended a powerful inverse to the classical “contraction principle.” Finally, Pukhalsky (1991) established the equivalence, under suitable regularity conditions, of the exponential tightness and the goodness of the rate function.Google Scholar
- Strassen (1964) established his formidable law of the iterated logarithm by direct estimates. A detailed exposition of the original approach appears in Freedman (1971b). Varadhan (1984) recognized the result as a corollary to Schilder’s theorem, and a complete proof along the suggested lines appears in Deuschel and Stroock (1989).Google Scholar
- Gentle introductions to large deviation theory and its applications are given by Varadhan (1984) and Dembo and Zeitouni (1998). The more demanding text of Deuschel and Stroock (1989) provides much additional insight to the persistent reader.Google Scholar

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