Large deviations

2006/2007, sem. 2

Prof. Boris Tsirelson (School of Mathematical Sciences).
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.)
Amir DEMBO, Ofer ZEITOUNI, "Large deviations techniques and applications", 1993.
Grading policy
Written homework and oral exam.

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