Large deviations. (English) Zbl 1228.60037

Varadhan, S. R. S.

Bhatia, Rajendra (ed.) et al., Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. I: Plenary lectures and ceremonies. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency (ISBN 978-981-4324-30-4/set; 978-81-85931-08-3/hbk; 978-981-4324-31-1/hbk; 978-981-4324-35-9/ebook). 622-639 (2011).

Summary: The theory of large deviations deals with techniques for estimating probabilities of rare events. These probabilities are exponentially small in a natural parameter and the task is to determine the exponential constant. To be precise, we have a family \(P_n\) of probability distributions on a space \({\mathcal X}\) and asymptotically
\[ P_n(A)= \exp\Big[ -n\inf_{x\in A}I(x)+o(n)\Big] \]
for a large class of sets, with a suitable choice of the function \(I(x)\). This function is almost always related to some form of entropy. There are connections to statistical mechanics as well as applications to the study of scaling limits for large systems. The subject had its origins in the Scandinavian insurance industry where it was used for the evaluation of risk. Since then, it has undergone many developments, and we review some of the recent progress.
For the entire collection see [Zbl 1220.00031].

MSC:

60F10	Large deviations
60-02	Research exposition (monographs, survey articles) pertaining to probability theory

Keywords:

large deviations