IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2208.02516.html
   My bibliography  Save this paper

Weak convergence to derivatives of fractional Brownian motion

Author

Listed:
  • S{o}ren Johansen
  • Morten {O}rregaard Nielsen

Abstract

It is well known that, under suitable regularity conditions, the normalized fractional process with fractional parameter $d$ converges weakly to fractional Brownian motion for $d>1/2$. We show that, for any non-negative integer $M$, derivatives of order $m=0,1,\dots,M$ of the normalized fractional process with respect to the fractional parameter $d$, jointly converge weakly to the corresponding derivatives of fractional Brownian motion. As an illustration we apply the results to the asymptotic distribution of the score vectors in the multifractional vector autoregressive model.

Suggested Citation

  • S{o}ren Johansen & Morten {O}rregaard Nielsen, 2022. "Weak convergence to derivatives of fractional Brownian motion," Papers 2208.02516, arXiv.org, revised Oct 2022.
    • Handle: RePEc:arx:papers:2208.02516
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2208.02516
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Søren Johansen & Morten Ørregaard Nielsen, 2012. "Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model," Econometrica, Econometric Society, vol. 80(6), pages 2667-2732, November.
    2. Franchi, Massimo, 2010. "A Representation Theory For Polynomial Cofractionality In Vector Autoregressive Models," Econometric Theory, Cambridge University Press, vol. 26(4), pages 1201-1217, August.
    3. Johansen, Søren & Nielsen, Morten Ørregaard, 2016. "The Role Of Initial Values In Conditional Sum-Of-Squares Estimation Of Nonstationary Fractional Time Series Models," Econometric Theory, Cambridge University Press, vol. 32(5), pages 1095-1139, October.
    4. Javier Hualde, 2012. "Weak convergence to a modified fractional Brownian motion," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(3), pages 519-529, May.
    5. Einmahl, Uwe, 1989. "Extensions of results of Komlós, Major, and Tusnády to the multivariate case," Journal of Multivariate Analysis, Elsevier, vol. 28(1), pages 20-68, January.
    6. Hualde, Javier, 2014. "Estimation of long-run parameters in unbalanced cointegration," Journal of Econometrics, Elsevier, vol. 178(2), pages 761-778.
    7. Marinucci, D. & Robinson, P. M., 2000. "Weak convergence of multivariate fractional processes," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 103-120, March.
    8. Johansen, SØren, 2008. "A Representation Theory For A Class Of Vector Autoregressive Models For Fractional Processes," Econometric Theory, Cambridge University Press, vol. 24(3), pages 651-676, 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. Javier Haulde & Morten Ørregaard Nielsen, 2022. "Fractional integration and cointegration," CREATES Research Papers 2022-02, Department of Economics and Business Economics, Aarhus University.
    2. Hualde, Javier, 2014. "Estimation of long-run parameters in unbalanced cointegration," Journal of Econometrics, Elsevier, vol. 178(2), pages 761-778.
    3. Sepideh Dolatabadi & Paresh Kumar Narayan & Morten Ørregaard Nielsen & Ke Xu, 2018. "Economic significance of commodity return forecasts from the fractionally cointegrated VAR model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 38(2), pages 219-242, February.
    4. Abbritti, Mirko & Carcel, Hector & Gil-Alana, Luis & Moreno, Antonio, 2023. "Term premium in a fractionally cointegrated yield curve," Journal of Banking & Finance, Elsevier, vol. 149(C).
    5. Stoupos, Nikolaos & Kiohos, Apostolos, 2022. "Euro area stock markets integration: Empirical evidence after the end of 2010 debt crisis," Finance Research Letters, Elsevier, vol. 46(PB).
    6. Filippo Beltrami & Fulvio Fontini & Monica Giulietti & Luigi Grossi, 2022. "The Zonal and Seasonal CO2 Marginal Emissions Factors for the Italian Power Market," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 83(2), pages 381-411, October.
    7. Ke Xu & Yu‐Lun Chen & Bo Liu & Jian Chen, 2024. "Price discovery and long‐memory property: Simulation and empirical evidence from the bitcoin market," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 44(4), pages 605-618, April.
    8. Yaya, OlaOluwa S. & Ogbonna, Ahamuefula E. & Olubusoye, Olusanya E., 2019. "How persistent and dynamic inter-dependent are pricing of Bitcoin to other cryptocurrencies before and after 2017/18 crash?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 531(C).
    9. Federico Carlini & Paolo Santucci de Magistris, 2019. "Resuscitating the co-fractional model of Granger (1986)," Discussion Papers 19/01, University of Nottingham, Granger Centre for Time Series Econometrics.
    10. Uwe Hassler & Marc-Oliver Pohle, 2019. "Forecasting under Long Memory and Nonstationarity," Papers 1910.08202, arXiv.org.
    11. Gilles de Truchis & Elena Ivona Dumitrescu & Florent Dubois, 2019. "Local Whittle Analysis of Stationary Unbalanced Fractional Cointegration Systems," EconomiX Working Papers 2019-15, University of Paris Nanterre, EconomiX.
    12. Søren Johansen & Morten Ørregaard Nielsen, 2019. "Nonstationary Cointegration in the Fractionally Cointegrated VAR Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 40(4), pages 519-543, July.
    13. Søren Johansen & Morten Ørregaard Nielsen, 2018. "Testing the CVAR in the Fractional CVAR Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(6), pages 836-849, November.
    14. Dolatabadi, Sepideh & Nielsen, Morten Ørregaard & Xu, Ke, 2016. "A fractionally cointegrated VAR model with deterministic trends and application to commodity futures markets," Journal of Empirical Finance, Elsevier, vol. 38(PB), pages 623-639.
    15. Massimo Franchi & Paolo Paruolo, 2019. "A general inversion theorem for cointegration," Econometric Reviews, Taylor & Francis Journals, vol. 38(10), pages 1176-1201, November.
    16. Federico Carlini & Paolo Santucci de Magistris, 2019. "On the Identification of Fractionally Cointegrated VAR Models With the Condition," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(1), pages 134-146, January.
    17. Morten Ørregaard Nielsen & Sergei S. Shibaev, 2015. "Forecasting daily political opinion polls using the fractionally cointegrated VAR model," Working Paper 1340, Economics Department, Queen's University.
    18. Stoupos, Nikolaos & Kiohos, Apostolos, 2021. "Energy commodities and advanced stock markets: A post-crisis approach," Resources Policy, Elsevier, vol. 70(C).
    19. Xu, Ke & Chen, Yu-Lun & Yang, J. Jimmy, 2023. "Market uncertainty, persistent arbitrage-free violation, and price discovery in RMB market," International Review of Financial Analysis, Elsevier, vol. 90(C).
    20. Alexander Boca Saravia & Gabriel Rodríguez, 2022. "Presidential approval in Peru: an empirical analysis using a fractionally cointegrated VAR," Economic Change and Restructuring, Springer, vol. 55(3), pages 1973-2010, August.

    More about this item

    Statistics

    Access and download statistics

    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:arx:papers:2208.02516. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.