Large deviations for Gaussian stationary processes and semi-classical analysis (original) (raw)

Large deviations for quadratic forms of stationary Gaussian processes

Stochastic Processes and their Applications, 1997

A large deviation principle is proved for Toeplitz quadratic forms of centred stationary Gaussian processes. The rate function is obtained by a sharp study of the behaviour of eigenvalues of a product of two Toeplitz matrices. Some statistical applications such as the likelihood ratio test and the estimation of the parameter of an autoregressive Gaussian process are also provided.

Large deviations for stationary Gaussian processes

Communications in Mathematical Physics, 1985

In their previous work on large deviations the authors always assumed the base process to be Markovian whereas here they consider the base process to be stationary Gaussian. Similar large deviation results are obtained under natural hypotheses on the spectral density function of the base process. A rather explicit formula for the entropy involved is also obtained.

Sharp large deviations for Gaussian quadratic forms with applications

ESAIM: Probability and Statistics, 2000

Under regularity assumptions, we establish a sharp large deviation principle for Hermitian quadratic forms of stationary Gaussian processes. Our result is similar to the well-known Bahadur-Rao theorem [2] on the sample mean. We also provide several examples of application such as the sharp large deviation properties of the Neyman-Pearson likelihood ratio test, of the sum of squares, of the Yule-Walker estimator of the parameter of a stable autoregressive Gaussian process, and finally of the empirical spectral repartition function.

Large deviations for quadratic functionals of Gaussian processes

Journal of Theoretical Probability, 1997

The Large Deviation Principle (LDP) is derived for several quadratic additive functionals of centered stationary Gaussian processes. For example, the rate function corresponding to 1/TST X2 dt is the Fenchel-Legendre transform of L(y)= –(1/4fl) Joo log(1 –4nyf(s)) ds, where Xt is a ...

On large deviations of empirical measures for stationary Gaussian processes

Stochastic Processes and their Applications, 1995

We show that the Large Deviation Principle with respect to the weak topology holds for the empirical measure of any stationary continuous time Gaussian process with continuous vanishing at innity spectral density. W e also point out that Large Deviation Principle might fail in both continuous and discrete time if the spectral density is discontinuous.

The Trace Problem for Toeplitz Matrices and Operators and its Impact in Probability

2013

The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by Grenander and Szeg\"o, "Toeplitz forms and their applications". It has then been extensively studied in the literature. In this paper we provide a survey and unified treatment of the trace approximation problem both for Toeplitz matrices and for operators and describe applications to discrete- and continuous-time stationary processes. The trace approximation problem serves indeed as a tool to study many probabilistic and statistical topics for stationary models. These include central and non-central limit theorems and large deviations of Toeplitz type random quadratic functionals, parametric and nonparametric estimation, prediction of the future value based on the observed past of the process, etc. We review and summarize the known results concerning the trace approximation problem, prove some new results, and provide a number of applica...

THE APPROXIMATE DENSITIES OF SOME QUADRATIC FORMS OF STATIONARY RANDOM VARIABLES

Journal of Time Series Analysis, 1987

First we obtain a convenient way of expressing the determinant of the difference between an identity matrix and some products of Toeplitz matrices. Then, using these results, we show that for a large number of normal processes there exist some quadratic forms whose matrices are of Toeplitz type such that their joint density admits an Edgeworth expansion.