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Representation and Weak Convergence of Stochastic Integrals with Fractional Integrator Processes

Author

Listed:
  • James Davidson

    (Department of Economics, University of Exeter)

  • Nigar Hashimzade

    (University of Reading)

Abstract

This paper considers the asymptotic distribution of the covariance of a nonstationary fractionally integrated process with the stationary increments of another such process - possibly, itself. Questions of interest include the relationship between the harmonic representation of these random variables, which we have analysed in a previous paper, and the construction derived from moving average representations in the time domain. The limiting integrals are shown to be expressible in terms of functionals of Itô integrals with respect to two distinct Brownian motions. Their mean is nonetheless shown to match that of the harmonic representation, and they satisfy the required integration by parts rule. The advantages of our approach over the harmonic analysis include the facts that our formulae are valid for the full range of the long memory parameters, and extend to non-Gaussian processes.

Suggested Citation

  • James Davidson & Nigar Hashimzade, 2008. "Representation and Weak Convergence of Stochastic Integrals with Fractional Integrator Processes," Discussion Papers 0807, University of Exeter, Department of Economics.
    • Handle: RePEc:exe:wpaper:0807
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    File URL: https://exetereconomics.github.io/RePEc/dpapers/DP0807.pdf
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    References listed on IDEAS

    as
    1. Bender, Christian, 2003. "An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter," Stochastic Processes and their Applications, Elsevier, vol. 104(1), pages 81-106, March.
    2. Vladas Pipiras & Murad S. Taqqu, 2002. "Deconvolution of fractional brownian motion," Journal of Time Series Analysis, Wiley Blackwell, vol. 23(4), pages 487-501, July.
    3. de Jong, Robert M. & Davidson, James, 2000. "The Functional Central Limit Theorem And Weak Convergence To Stochastic Integrals I," Econometric Theory, Cambridge University Press, vol. 16(5), pages 621-642, October.
    4. Davidson, James & Hashimzade, Nigar, 2008. "Alternative Frequency And Time Domain Versions Of Fractional Brownian Motion," Econometric Theory, Cambridge University Press, vol. 24(1), pages 256-293, February.
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    Cited by:

    1. Buchmann, Boris & Chan, Ngai Hang, 2013. "Unified asymptotic theory for nearly unstable AR(p) processes," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 952-985.
    2. Bent Jesper Christensen & Robinson Kruse & Philipp Sibbertsen, 2013. "A unified framework for testing in the linear regression model under unknown order of fractional integration," CREATES Research Papers 2013-35, Department of Economics and Business Economics, Aarhus University.

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    More about this item

    Keywords

    Stochastic integral; weak convergence; fractional Brownian motion.;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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