# Level-3 Large Deviations for I.I.D. Random Vectors

Chapter

## Abstract

Theorem II.4.4 stated the level-3 large deviation property for i.i.d. random vectors taking values in ℝ^{ d }. In this chapter, we prove Theorem II.4.4 in the special case of i.i.d. random variables with a finite state space. This version of the theorem covers the applications of level-3 large deviations which were made in Chapters III, IV, and V to the Gibbs variational principle. Theorem II.4.4 can also be proved via the methods of Donsker and Varadhan (1983a). The main result in that paper is a level-3 theorem for continuous parameter Markov processes taking values in a complete separable metric space.^{1}

## Keywords

Random Vector Relative Entropy Entropy Function Borel Probability Measure Contraction Principle
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## Copyright information

© Springer-Verlag New York Inc. 1985