Sharp large deviations for Gaussian quadratic forms with applications

Abstract

Abstract.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.Résumé. Sous des hypothèses de régularité convenables, onétablit un principe de grandes déviations précises pour des formes quadratiques de processus gaussiens stationnaires. Notre résultat est l'analogue du théorème de Bahadur-Rao [2] sur la moyenne empirique. Nous proposonségale-ment plusieurs exemples d'application comme les propriétés de grandes déviations précises pour le test du rapport de vraisemblance de Neyman-Pearson, pour la somme des carrés, pour l'estimateur de Yule-Walker du paramètre d'un processus gaussien autorégressif stable, et finalement pour la fonction de répartition spectrale empirique.

Nonparametric signal detection with small type I and type II error probabilities

“…The study of the asymptotic of moderate and large deviation probabilities for quadratic forms of Gaussian random variables is a rather popular research subject (Kakizawa 2007;Bercue et al 1997;Bercu et al 2000;Bryc and Dembo 1997). Our reasoning is based on the approach proposed in Bercu et al (2000).…”

“…Instead of the CLT we establish some versions of the Cramer Theorem. The proofs of these versions require some modification of techniques of previous papers (see Bercue et al 1997;Bercu et al 2000;Bryc and Dembo 1997;Kakizawa 2007) devoted to moderate deviation probabilities of quadratic forms of Gaussian random variables. To some extent we implement the methods of Ermakov (2008) as well.…”

“…Our reasoning is based on the approach proposed in Bercu et al (2000). However the technique of Toeplitz matrices developed in Bercu et al (2000) could not be implemented for this setup. This causes the differences in the proof.…”

Two Examples of Non Strictly Convex Large Deviations

“…for all x ∈ K M , for some (smooth) strictly positive function C. This analysis is based on the so-called theory of sharp large deviations, developed in [4,5], and used in [3,20,21] for diffusion processes and statistical estimators thereof. It is based on refinements of the Gärtner-Ellis theorem in the case where the limiting cgf is not steep at the boundary; these refinements, using a time-dependent change of measure, were introduced in [7] and [11].…”

Sharp Large Deviation for the Energy of -Brownian Bridge

“…Recently, the sharp large deviation principle is widely used in the study of Gaussian quadratic forms, Ornstein-Uhlenbeck model, and fractional OrnsteinUhlenbeck (cf. Bercu and Rouault [7], Bercu et al [8], and Bercu et al [9,10]). …”