### Large deviations

#### 2006/2007, sem. 2

Lecturer
Prof. Boris Tsirelson (School of Mathematical Sciences).
Time and place
Thursday 17-20 Schreiber 209.
Prerequisites
Be acquainted with such things as compact metric spaces and the Hilbert space L2 of square integrable functions on a measure space. Everything else will be explained from scratch. However, some maturity in analysis is needed. (Maturity in probability is not needed.)
Textbook
Amir DEMBO, Ofer ZEITOUNI, "Large deviations techniques and applications", 1993.
Written homework and oral exam.

### Syllabus (tentative)

Basic notions: large deviation principle (LDP), rate function, contraction principle, change of measure.

Binomial and multinomial LDP: Sanov's theorem. Relative entropy as a rate function.

LDP for random variables: Cramer's theorem via Gibbs's conditioninig.

Infinite dimension as limit of finite dimensions: the Dawson-Gartner theorem.

LDP for random functions: Mogulskii's theorem.

Random walks with drift: The Freidlin-Wentzell theory.

Moderate deviations.

#### Good news:

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2007 to Srinivasa S. R. Varadhan. Courant Institute of Mathematical Sciences, New York for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations.
Srinivasa S. R. Varadhan was born January 2, 1940 in Madras (Chennai), India. He received his B.Sc. honours degree in 1959 and his M.A. the following year, both from Madras University. In 1963 he received his Ph.D. from the Indian Statistical Institute. He is currently Professor of Mathematics and Frank J. Gould Professor of Science at the Courant Institute.
The Abel Prize is an international prize for outstanding scientific work in the field of mathematics. The prize is meant to recognize contributions of extraordinary depth and influence to the mathematical sciences.

#### Three quotes:

"...a particularly convenient way of stating asymptotic results that, on the one hand, are accurate enough to be useful and, on the other hand, are loose enough to be correct."

Amir DEMBO, Ofer ZEITOUNI, "Large deviations techniques and applications",
Jones and Bartlett Publ., 1993 (page 5).

"The most fascinating aspect of the theory is that the exponential decay rates are computable in terms of entropy functions."

Richard S. ELLIS, "Entropy, large deviations, and statistical mechanics",
Springer, 1985 (page vii).

"First, there is a huge potential for applying these ideas to the study, analysis, and design of stochastic systems -- in particular, information systems. Second, the subject material is unusually technical."

James A. BUCKLEW, "Large deviations techniques in decision, simulation, and estimation",
Wiley, 1990 (page vii).

### Lecture notes

1. Physical prelude.
PDF or Postscript.
2. Basic notions.
PDF or Postscript.
3. Entropy appears.
PDF or Postscript.
4. More on the basic notions.
PDF or Postscript.
5. LDP in spaces of functions.
PDF or Postscript.
6. Physics, entropy, and large deviations.
PDF or Postscript.
7. Against the stream: the Freidlin-Wentzell theory.
PDF or Postscript.
8. Blocks, Markov chains, Ising model.
PDF or Postscript.
9. Beyond compactness: basic notions (revised 11.06.07).
PDF or Postscript.
10. Cramer theorem via Sanov's theorem.
PDF or Postscript.