Large deviations for quadratic forms of stationary Gaussian processes (original) (raw)

Large deviations for Gaussian stationary processes and semi-classical analysis

In this paper, we obtain a large deviation principle for quadratic forms of Gaussian stationary processes. It is established by the conjunction of a result of Roch and Silbermann on the spectrum of products of Toeplitz matrices together with the analysis of large deviations carried out by Gamboa, Rouault and the first author. An alternative proof of the needed result on Toeplitz matrices, based on semi-classical analysis, is also provided. Keywords Large deviations • Gaussian processes • Toeplitz matrices • Distribution of eigenvalues Pn(x 0 , x 1 , x 2 ,. . .

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 ...

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.

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 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.

On the Central Limit Theorem for Toeplitz Quadratic Forms

2005

Let X(t), t = 0, ±1,. .. , be a real-valued stationary Gaussian sequence with a spectral density function f (λ). The paper considers the question of applicability of the central limit theorem (CLT) for a Toeplitz-type quadratic form Qn in variables X(t), generated by an integrable even function g(λ). Assuming that f (λ) and g(λ) are regularly varying at λ = 0 of orders α and β, respectively, we prove the CLT for the standard normalized quadratic form Qn in a critical case α + β = 1 2. We also show that the CLT is not valid under the single condition that the asymptotic variance of Qn is separated from zero and infinity.