Rate functions in the theory of large deviations (Q1914314)

Let \(X\) be a normal topological space and \((P_n)_n\) a sequence of probability measures on \(X\) which satisfies a large deviation principle on \(X\) with a rate function \(I\). \textit{I. H. Dinwoodie} [Ann. Probab. 21, No. 1, 216-231 (1993; Zbl 0777.60024)] has established that \(I\) can be represented as \[ I(x) = \sup \bigl\{ f(x) - \Phi (f),\;f \in C_b (X) \bigr\}, \] where \(C_b(X)\) denotes the space of all bounded continuous functions and \[ \Phi (f) : = \lim_{n \to \infty} {1 \over n} \log \int_X \exp \bigl( nf(x) \bigr) dP_n (x). \] The author proves an analog representation of \(I\) with \(C_b(X)\) replaced by a suitable subclass of continuous functions using Varadhan's integral lemma only. One application is the identification of the Cramér rate in a separable Banach space.