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Large Deviations for Quadratic Functionals of Gaussian Processes

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  • Włodzimierz Bryc
  • Amir Dembo

Abstract

The Large Deviation Principle (LDP) is derived for several quadratic additive functionals of centered stationary Gaussian processes. For example, the rate function corresponding to $$1/T\int {_0^T } X_t^2 dt$$ is the Fenchel-Legendre transform of $$L(y) = - (1/4\pi )\int {_{ - \infty }^\infty } \log (1 - 4\pi yf(s))ds$$ where X t is a continuous time process with the bounded spectral density f(s). This spectral density condition is strictly weaker than the one necessary for the LDP to hold for all bounded continuous functionals. Similar results are obtained for the energy of multivariate discrete-time Gaussian processes and in the regime of moderate deviations, the latter yielding the corresponding Central Limit Theorems.

Suggested Citation

  • Włodzimierz Bryc & Amir Dembo, 1997. "Large Deviations for Quadratic Functionals of Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 10(2), pages 307-332, April.
    • Handle: RePEc:spr:jotpro:v:10:y:1997:i:2:d:10.1023_a:1022656331883
    DOI: 10.1023/A:1022656331883
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    References listed on IDEAS

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    1. Bryc, Wlodzimierz & Smolenski, Wlodzimierz, 1993. "On the large deviation principle for a quadratic functional of the autoregressive process," Statistics & Probability Letters, Elsevier, vol. 17(4), pages 281-285, July.
    2. Bryc, Wlodzimierz, 1993. "A remark on the connection between the large deviation principle and the central limit theorem," Statistics & Probability Letters, Elsevier, vol. 18(4), pages 253-256, November.
    3. Bryc, Wlodzimierz & Dembo, Amir, 1995. "On large deviations of empirical measures for stationary Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 58(1), pages 23-34, July.
    4. Dembo, Amir & Zajic, Tim, 1995. "Large deviations: From empirical mean and measure to partial sums process," Stochastic Processes and their Applications, Elsevier, vol. 57(2), pages 191-224, June.
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