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

Donsker, M. D.; Varadhan, S. R. S.

Commun. Math. Phys. 97, No. 1-2, 187-210 (1985).

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

MSC:

60F10	Large deviations
60G10	Stationary stochastic processes

Keywords:

large deviation principle for stationary ergodic Gaussian sequences; stationary measures on the path space; entropy function; spectral density

References:

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[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) · 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
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