Large deviations for stationary Gaussian processes. (English) Zbl 0646.60030

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.
Reviewer: E.Bolthausen


60F10 Large deviations
60G10 Stationary stochastic processes
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[1] Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Wiener integrals for large time. Functional Integration and its Applications, Proceedings of the International Conference held at the Cumberland Lodge, Windsor Great Park, London, in April 1974, A. M. Arthurs, (ed.). Oxford: Clarendon 1975
[2] Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time, I. Commun. Pure Appl. Math.28, 1-47 (1976), II. Commun. Pure Appl. Math.28, 279-301 (1975); III. Commun. Pure Appl. Math.29, 389-461 (1976); IV. Commun. Pure Appl. Math.36, 183-212 (1983) · Zbl 0323.60069 · doi:10.1002/cpa.3160280102
[3] Donsker, M.D., Varadhan, S.R.S.: Asymptotics for the Wiener Sausage. Commun. Pure Appl. Math.28, 525-565 (1975) · Zbl 0333.60077 · doi:10.1002/cpa.3160280406
[4] Donsker, M.D., Varadhan, S.R.S.: On laws of the iterated logarithm for local times. Commun. Pure Appl. Math.30, 707-753 (1977) · Zbl 0367.60095 · doi:10.1002/cpa.3160300603
[5] Donsker, M.D., Varadhan, S.R.S.: Asymptotics for the Polaron. Commun. Pure Appl. Math.36, 505-528 (1983) · Zbl 0538.60081 · doi:10.1002/cpa.3160360408
[6] Orey, S.: Large deviations and Shanon-McMillan theorems (preprint)
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