# large deviation theory

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

as grows large, where 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