Large and Moderate Deviations for Projective Systems and Projective Limits

Tryfon Daras

doi:10.4236/am.2012.312a282

Abstract

One of the active fields in applied probability, the last two decades, is that of large deviations theory i.e. the one dealing with the (asymptotic) computation of probabilities of rare events which are exponentially small as a function of some parameter e.g. the amplitude of the noise perturbing a dynamical system. Basic ideas of the theory can be tracked back to Laplace, the first rigorous results are due to Cramer although a clear definition was introduced by Varadhan in 1966. Large deviations estimates have been proved to be the crucial tool in studying problems in Statistics, Physics (Thermodynamics and Statistical Mechanics), Finance (Monte-Carlo methods, option pricing, long term portfolio investment) and in Applied probability (queuing theory). The aim of this work is to describe one of the (recent) methods of proving large deviations results, namely that of projective systems. We compare the method with the one of projective limits and show the advantages of the first. These advantages are due to the fact that: 1) the arguments are direct and the proofs of the basic results of the theory are much easier and simpler; 2) we are able to extend most of these results using suitable projective systems. We apply the method in the case of a) sequences of i.i.d. r.v.’s and b) sequences of exchangeable r.v.’s. All the results are being proved in a simple “unified” way.

Highlights

One of the active fields in applied probability, the last two decades, is that of large deviations theory i.e. the one dealing with the computation of probabilities of rare events which are exponentially small as a function of some parameter e.g. the amplitude of the noise perturbing a dynamical system
If the upper bound is valid for all compact sets, while the lower bound is still true for all open sets, we say that the net of p.m.’s viations principle
When someone deals with the empirical measures of an i.i.d sequence, the following large deviations result is true

Summary

Notation and Basic Results

The following two theorems give large deviations results in the case of projective systems [3]. Let and assume that: be a net of p.m.’s on F i) A , the net of p.m.’s d 1 d D satisfies a large deviations principle with normalizing constants r d d D and rate function. The following basic result, analogous to that of Theorem 1.7, allows one to transport a large deviations result on a “smaller” topological space to a “larger” one. Assume that A , the net of p.m.’s d 1 d D satisfies the full large deviations principle with constants r d d D and good rate function. The motivation for this paper was to find a “unified” way of proving large deviations results This is done by using the projective systems approach. We are able to prove extensions of these results to more abstract spaces, at least in the case of exchangeable sequences of r.v.’s

Applications

F A the probability measure: