Large and moderate deviations in probability theory

2014/2015, sem. 2

Prof. Boris Tsirelson (School of Mathematical Sciences).
Be acquainted with Lebesgue integration and metric spaces. Everything else will be explained from scratch. However, some maturity in analysis is needed.
Amir DEMBO, Ofer ZEITOUNI, "Large deviations techniques and applications", 1993.
Grading policy
Written homework and oral exam.

Lecture notes

  1. Introduction: within and beyond the normal approximation.
  2. Cramer's theorem.
  3. The least unlikely way maximizes entropy.
  4. Large deviations in spaces of functions.
  5. Moderate deviations in spaces of functions.
  6. Small random perturbations of deterministic dynamics.

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).