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large deviation theory

January 3, 2010

Large deviation theory is the study of random variables X_n whose probability densities follow a decaying exponential of the form

p(x) \approx e^{- n I(x) }

as n grows large, where I(x) is known as the rate-function.  This naturally leads to a generalization of both the law of large numbers and the central limit theorem, and forms a natural framework upon which a rigorous theory of equilibrium statistical mechanics can be built.  In addition, large deviation theory has powerful applications even to traditionally “non-equilibrium” situations.  This paper, which was written as part of the senior comps requirement for my math major, provides a basic (although rigorous) introduction to large deviation theory, as well as its application to a simple class of stochastic differential equations.  It should be accessible to anyone with an introductory course in analysis. PDF

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