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Limit theorems for power variations of ambit fields driven by white noise

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  • Pakkanen, Mikko S.

Abstract

We study the asymptotics of lattice power variations of two-parameter ambit fields driven by white noise. Our first result is a law of large numbers for power variations. Under a constraint on the memory of the ambit field, normalized power variations converge to certain integral functionals of the volatility field associated to the ambit field, when the lattice spacing tends to zero. This result holds also for thinned power variations that are computed by only including increments that are separated by gaps with a particular asymptotic behavior. Our second result is a stable central limit theorem for thinned power variations.

Suggested Citation

  • Pakkanen, Mikko S., 2014. "Limit theorems for power variations of ambit fields driven by white noise," Stochastic Processes and their Applications, Elsevier, vol. 124(5), pages 1942-1973.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:5:p:1942-1973
    DOI: 10.1016/j.spa.2014.01.005
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    References listed on IDEAS

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    1. Réveillac, Anthony, 2009. "Estimation of quadratic variation for two-parameter diffusions," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1652-1672, May.
    2. Gabriel Lang & François Roueff, 2001. "Semi-parametric Estimation of the Hölder Exponent of a Stationary Gaussian Process with Minimax Rates," Statistical Inference for Stochastic Processes, Springer, vol. 4(3), pages 283-306, October.
    3. Ole E. Barndorff–Nielsen & Fred Espen Benth & Almut E. D. Veraart, 2010. "Ambit processes and stochastic partial differential equations," CREATES Research Papers 2010-17, Department of Economics and Business Economics, Aarhus University.
    4. Corcuera, José Manuel & Hedevang, Emil & Pakkanen, Mikko S. & Podolskij, Mark, 2013. "Asymptotic theory for Brownian semi-stationary processes with application to turbulence," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2552-2574.
    5. Cheridito, Patrick, 2004. "Gaussian moving averages, semimartingales and option pricing," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 47-68, January.
    6. Soulier, Philippe, 2001. "Moment bounds and central limit theorem for functions of Gaussian vectors," Statistics & Probability Letters, Elsevier, vol. 54(2), pages 193-203, September.
    7. Ole E. Barndorff–Nielsen & Fred Espen Benth & Almut E. D. Veraart, 2010. "Modelling electricity forward markets by ambit fields," CREATES Research Papers 2010-41, Department of Economics and Business Economics, Aarhus University.
    8. Barndorff-Nielsen, Ole E. & Corcuera, José Manuel & Podolskij, Mark, 2009. "Power variation for Gaussian processes with stationary increments," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1845-1865, June.
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    Cited by:

    1. Ole E. Barndorff-Nielsen & Orimar Sauri & Benedykt Szozda, 2017. "Selfdecomposable Fields," Journal of Theoretical Probability, Springer, vol. 30(1), pages 233-267, March.
    2. Mikko S. Pakkanen & Anthony Réveillac, 2014. "Functional limit theorems for generalized variations of the fractional Brownian sheet," CREATES Research Papers 2014-14, Department of Economics and Business Economics, Aarhus University.
    3. Mark Podolskij & Nopporn Thamrongrat, 2015. "A weak limit theorem for numerical approximation of Brownian semi-stationary processes," CREATES Research Papers 2015-53, Department of Economics and Business Economics, Aarhus University.
    4. Mark Podolskij, 2014. "Ambit fields: survey and new challenges," CREATES Research Papers 2014-51, Department of Economics and Business Economics, Aarhus University.

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