Large deviations for stationary Gaussian processes
type: Item
instance of: scholarly article
title: Large deviations for stationary Gaussian processes
zbMATH Open document ID: 0646.60030
DOI: 10.1007/BF01206186
author: S. R. S. Varadhan
author: M. D. Donsker
published in: Communications in Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: The authors derive a so-called level 3 large deviation principle for stationary ergodic Gaussian sequences \(X_ k\), \(k\in {\mathbb{Z}}:\) For \(\omega =\{x_ k\}\) in the path space \({\mathbb{R}}^{{\mathbb{Z}}}\), let \(\omega^{(n)}\) be the sequence made periodic with period n and agreeing with \(\omega\) at coordinates 1 to n. Then \(\pi_ n(\omega)\) is defined by \[ \pi_ n(\omega)=n^{-1}\sum^{n-1}_{j=0}\delta_{T\quad j\omega}(n) \] where T j is the usual shift. Then for a Borel subset A in the space of stationary measures on the path space \[ -\inf_{R\in int(A)}H(R)\leq \overline{\lim}\overline{_{n\to \infty}}n^{-1} \log P(\pi \quad_ n\in A)\leq -\inf_{R\in cl(A)}H(R) \] where inf and cl refer to the weak topology. The entropy function H can be given explicitly in terms of the spectral density.
reviewed by: Erwin Bolthausen
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 4054700
zbMATH Keywords: large deviation principle for stationary ergodic Gaussian sequences
zbMATH Keywords: stationary measures on the path space
zbMATH Keywords: entropy function
zbMATH Keywords: spectral density
MaRDI profile type: Publication
cites work: Asymptotic evaluation of certain markov process expectations for large time, I
cites work: Asymptotics for the wiener sausage
cites work: Asymptotics for the polaron
description: scientific article; zbMATH DE number 4054700