Exponential Tightness and Projective Systems in Large Deviation Theory

Abstract

Let {E α ,p α β } be a projective system of Hausdorff topological spaces; here α,β∈A, a directed set, and p α β :E β →E α is a continuous surjective map for α<β (for details, see Section 3). Let E be a Hausdorff topological space endowed with a σ-algebra,e (possibly smaller than the Borel σ-algebra), and for each α∈A, let p α :E→E α be a continuous measurable surjective map such that p α =p α β 0p β for a α<β Let {µn}be a sequence of probability measures on e,and assume that for each a α∈ A,the sequence {µn o p α -1 } satisfies the large deviation principle (see Section 2). In this paper we show that under suitable additional assumptions, the large deviation principle for {µn} follows.