Large Deviation Theory and Saddlepoint Approximation for Stochastic Processes

Abstract

A large number of results in Chapters 1–7 have been obtained via the central limit theorem (CLT) and the Edgeworth expansion of suitably normalized quantities. The latter is the basic tool for higher order asymptotic theory as in Chapter 4. On the other hand, there are other important tools, namely large deviation theorem and saddlepoint approximation. Section 8.1 is devoted to a derivation of a variant of the large deviation theorem, which turns out to be available for several statistical problems in Gaussian stationary processes, especially for quadratic functionals. Further we also give another example of the Ornstein-Uhlenbeck (O-U) diffusion process. In Section 8.2 we review general concepts of Bahadur’s asymptotic efficiency of estimators and tests based on the large deviation approach, with special emphasis on the spectral analysis of Gaussian stationary processes. Section 8.3 describes some developments for the O-U diffusion process.