Exponential Tightness and Projective Systems in Large Deviation Theory

Abstract

Let {E _α ,p ^β _α } be a projective system of Hausdorff topological spaces; here α,β∈A, a directed set, and p ^β _α :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,ε (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 ^β _α 0p _β for a α<β Let {µ_n}be a sequence of probability measures on ε,and assume that for each a α∈ A,the sequence {µ_n o p ^-1_α } satisfies the large deviation principle (see Section 2). In this paper we show that under suitable additional assumptions, the large deviation principle for {µ_n} follows.