## TAU:0366-5013 |
## Large and moderate deviations in probability theory | ## 2014/2015, sem. 2 |

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

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

"...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).