An Introduction to the Theory of Large Deviations

1. Front Matter

   Pages i-vii

   PDF

2. Introduction

   • D. W. Stroock

   Pages 1-2

3. Brownian Motion in Small Time, Strassen’s Iterated Logarithm

   • D. W. Stroock

   Pages 2-22

4. Large Deviations, Some Generalities

   • D. W. Stroock

   Pages 23-29

5. Cramér’s Theorem

   • D. W. Stroock

   Pages 30-75

6. Large Deviation Principle for Diffusions

   • D. W. Stroock

   Pages 75-101

7. Introduction to Large Deviations from Ergodic Phenomena

   • D. W. Stroock

   Pages 101-114

8. Existence of a Rate Function

   • D. W. Stroock

   Pages 114-131

9. Identification of the Rate Function

   • D. W. Stroock

   Pages 131-155

10. Some Non-Uniform Large Deviation Results

    • D. W. Stroock

    Pages 155-179

11. Logarithmic Sobolev Inequalities

    • D. W. Stroock

    Pages 179-195

12. Back Matter

    Pages 196-196

    PDF

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 time'when 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 Azencott's 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).

• Brownian motion
• Deviations
• Excel
• Grosse Abweichung
• Large
• attribute
• boundary element method
• form
• identification
• language
• logarithm
• mathematics
• organization
• set
• theorem

• Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA

  D. W. Stroock

Policies and ethics