ABSTRACT
We followed here largely the logical structure of [DeZ98]. That is, Section 6.2 describes the definition of the large deviation principle (LDP) and some of its equivalent formulations and basic properties. Section 6.3 provides an overview of large deviation theorems in H . Moving to a more abstract setup where the underlying variables take values in a topological space, Section 6.4 presents, after a short discussion on properties of the LDP, a collection of methods aimed at establishing the LDP. These methods include transformations of the LDP (i.e., how the LDP behaves under maps between spaces), relations between the LDP and Laplace's method for the evaluation for exponential integrals, properties of the LDP in topological vector spaces, and the behavior of the LDP under projective limits. Section 6.5 deals with LDPs for the sample paths of certain stochastic processes and the application of such LDPto the problem of the exit of randomly perturbed solutions of differential equations from the domain of attraction of stable equilibria. Section 6.6 deals with LDPs for the empirical measure of (discrete time) random processes: Sanov's theorem for the empirical measure of an i.i.d. sample and its extensions to Markov processes and mixing sequences are discussed. The section ends with two particular applications of the LDP: one to hypothesis testing problems in statistics, the other to the Gibbs conditioning principle in statistical mechanics.
