Moderate deviations for quadratic forms in Gaussian stationary processes

Abstract

Moderate deviations limit theorem is proved for quadratic forms in zero-mean Gaussian stationary processes. Two particular cases are the cumulative periodogram and the kernel spectral density estimator. We also derive the exponential decay of moderate deviation probabilities of goodness-of-fit tests for the spectral density and then discuss intermediate asymptotic efficiencies of tests.

Covariance matrix estimation for stationary time series

“…In fact, our result is more relevant to the so-called moderate deviations according to the terminology of Dembo and Zeitouni (1998). Bryc and Dembo (1997) and Kakizawa (2007) obtained moderate deviation principles for quadratic forms of Gaussian processes. Djellout, Guillin and Wu (2006) studied moderate deviations of periodograms of linear processes.…”

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

“…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.…”

“…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).…”

“…The large and moderate deviation probabilities of test statistics are the traditional research subject (see Nikitin 1995;Kakizawa 2007;Stute 1997 and references therein) in parametric and nonparametric hypothesis testing. Recently this problem has become treated for testing of hypothesis versus nonparametric sets of alternatives (see Inglot et al 1997;Suslina 2004, 2005;Ermakov 2008).…”

Large deviations for quadratic functionals of stable Gauss-Markov chains and entropy production

“…The asymptotic cumulant generating function does not contain apparently the whole information on the large deviation property of the process: there is a loss of information passing to the limit. For Toeplitz quadratic forms of stationary centered Gaussian sequences, large deviation principles are now well-established [29][30][31], as well as some moderate deviation principle [32]. These results have been obtained by a sharp study of the spectrum of a product of two Toeplitz matrices.…”