Two Books on Large Deviations

Large Deviations for Two Scaled Diffusions

Mathematical Subject Classification (1991) 60F10 Summary. We formulate large deviations principle (LDP) for diffusion pair (X, ξ) = (X t , ξ ε t ), where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time. More exactly, the LDP is established for (X, ν) with ν(dt, dz) being an occupation type measure corresponding to ξ t . In some sense ...

On large deviations mixing sequences

There is an extensive literature on the large deviation principle under various dependence structures. However, in comparison to other limit theorems, few papers deal with strong mixing dependence conditions. In this note we prove the large deviation principle for 4-mixing stationary random sequences with fast enough convergence rate and for $-mixing stationary random sequences. Our d-mixing re...

Two Recent Books on Algebras

theory, the relation of hypercomplex numbers to quadric surfaces, abelian integrals, etc. Moreover, as will be seen below, the book would have gained both in power and charm if the second part had been compressed. Part I, on the general theory of number fields (130 pages) is divided into three chapters. Of these, the first gives the definition of a number field via four postulates, the rudiment...

Large Deviations

Large Deviations

This paper is based on Wald Lectures given at the annual meeting of the IMS in Minneapolis during August 2005. It is a survey of the theory of large deviations. 1. Large deviations for sums. The role of " large deviations " is best understood through an example. Suppose that X random variables, for instance, normally distributed with mean zero and variance 1. Then, E e θ(X 1 +···+X n) = E[e θX ...