An Introduction to the Theory of Large Deviations These notes are based on a course which I gave during the academic year 1983 84 at the University of Colorado. My intention was to provide both my audience as well as myself with an introduction to the theory of 1arie deviations • The organization of sections 1) through 3) owes something to chance and a great deal to the excellent set of notes written by R. Azencott for the course which he gave in 1978 at Saint Flour (cf. Springer Lecture Notes in Mathematics 774). To be more precise: it is chance that I was around N. Y. U. at the timewhen M. Schilder wrote his thesis. and so it may be considered chance that I chose to use his result as a jumping off point; with only minor variations. everything else in these sections is taken from Azencott. In particular. section 3) is little more than a rewrite of his exoposition of the Cramer theory via the ideas of Bahadur and Zabel. Furthermore. the brief treatment which I have given to the Ventsel Freidlin theory in section 4) is again based on Azencotts ideas. All in all. the biggest difference between his and my exposition of these topics is the language in which we have written. However. another major difference must be mentioned: his bibliography is extensive and constitutes a fine introduction to the available literature. mine shares neither of these attributes. Starting with section 5). 0. Introduction 1. Brownian Motion in Small Time Strassen’s Iterated Logarithm 2. Large Deviations Some Generalities 3. Cramér’s Theorem 4. Large Deviation Principle for Diffusions 5. Introduction to Large Deviations from Ergodic Phenomena 6. Existence of a Rate Function 7. Identification of the Rate Function 8. Some Non Uniform Large Deviation Results 9. Logarithmic Sobolev Inequalities.