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    Hualde, Javier 2012. Weak convergence to a modified fractional Brownian motion. Journal of Time Series Analysis, Vol. 33, Issue. 3, p. 519.

    Davidson, James and Hashimzade, Nigar 2009. Type I and type II fractional Brownian motions: A reconsideration. Computational Statistics & Data Analysis, Vol. 53, Issue. 6, p. 2089.

    Lavancier, Frédéric Philippe, Anne and Surgailis, Donatas 2009. Covariance function of vector self-similar processes. Statistics & Probability Letters, Vol. 79, Issue. 23, p. 2415.



  • James Davidson (a1) and Nigar Hashimzade (a1)
  • DOI:
  • Published online: 01 February 2008

This paper compares models of fractional processes and associated weak convergence results based on moving average representations in the time domain with spectral representations. Both approaches have been applied in the literature on fractional processes. We point out that the conventional forms of these models are not equivalent, as is commonly assumed, even under a Gaussianity assumption. We show that it is necessary to distinguish between “two-sided” processes depending on both leads and lags from one-sided or “causal” processes, because in the case of fractional processes these models yield different limiting properties. We derive new representations of fractional Brownian motion and show how different results are obtained for, in particular, the distribution of stochastic integrals in the multivariate context. Our results have implications for valid statistical inference in fractional integration and cointegration models.We thank F. Hashimzade and two anonymous referees for their valuable comments.

Corresponding author
Address correspondence to James Davidson, School of Business and Economics, University of Exeter, Exeter EX4 4PU, U.K.; e-mail:
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Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
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