IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v10y1997i2d10.1023_a1022656331883.html
   My bibliography  Save this article

Large Deviations for Quadratic Functionals of Gaussian Processes

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1022656331883
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1022656331883?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Djellout, Hacène & Guillin, Arnaud & Samoura, Yacouba, 2017. "Estimation of the realized (co-)volatility vector: Large deviations approach," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 2926-2960.
    2. Heck, Matthias K., 1998. "The principle of large deviations for the almost everywhere central limit theorem," Stochastic Processes and their Applications, Elsevier, vol. 76(1), pages 61-75, August.
    3. Ken Duffy & David Malone, 2008. "Logarithmic asymptotics for a single-server processing distinguishable sources," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(3), pages 509-537, December.
    4. Pakdaman, Khashayar & Thieullen, Michèle & Wainrib, Gilles, 2012. "Asymptotic expansion and central limit theorem for multiscale piecewise-deterministic Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 122(6), pages 2292-2318.
    5. Zajic, Tim, 1996. "Large deviations for a class of recursive algorithms," Statistics & Probability Letters, Elsevier, vol. 26(2), pages 135-140, February.
    6. Duffy, Ken & Lobunets, Olena & Suhov, Yuri, 2007. "Loss aversion, large deviation preferences and optimal portfolio weights for some classes of return processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 378(2), pages 408-422.
    7. Xin Guo & Zhao Ruan & Lingjiong Zhu, 2015. "Dynamics of Order Positions and Related Queues in a Limit Order Book," Papers 1505.04810, arXiv.org, revised Oct 2015.
    8. Bercu, B. & Gamboa, F. & Rouault, A., 1997. "Large deviations for quadratic forms of stationary Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 71(1), pages 75-90, October.
    9. Federico Camerlenghi & Claudio Macci & Elena Villa, 2021. "Asymptotic behavior of mean density estimators based on a single observation: the Boolean model case," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(5), pages 1011-1035, October.
    10. Camilla Calì & Maria Longobardi & Claudio Macci & Barbara Pacchiarotti, 2022. "Asymptotic results for linear combinations of spacings generated by i.i.d. exponential random variables," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(6), pages 733-752, August.
    11. Beghin, Luisa & Macci, Claudio, 2017. "Asymptotic results for a multivariate version of the alternative fractional Poisson process," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 260-268.
    12. Dembo, Amir & Zeitouni, Ofer, 1996. "Large deviations for subsampling from individual sequences," Statistics & Probability Letters, Elsevier, vol. 27(3), pages 201-205, April.
    13. Dembo, Amir & Zajic, Tim, 1997. "Uniform large and moderate deviations for functional empirical processes," Stochastic Processes and their Applications, Elsevier, vol. 67(2), pages 195-211, May.
    14. Vijay G. Subramanian, 2010. "Large Deviations of Max-Weight Scheduling Policies on Convex Rate Regions," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 881-910, November.
    15. Eichelsbacher, Peter, 1998. "Large deviations in partial sums of U-processes in stronger topologies," Statistics & Probability Letters, Elsevier, vol. 38(3), pages 207-214, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:10:y:1997:i:2:d:10.1023_a:1022656331883. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.