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Asymptotic Theory for Fractional Regression Models via Malliavin Calculus

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  • Solesne Bourguin

    (Université de Paris 1 Panthéon-Sorbonne)

  • Ciprian A. Tudor

    (Université de Lille 1)

Abstract

We study the asymptotic behavior as n→∞ of the sequence $$S_{n}=\sum_{i=0}^{n-1}K\bigl(n^{\alpha}B^{H_{1}}_{i}\bigr)\bigl(B^{H_{2}}_{i+1}-B^{H_{2}}_{i}\bigr)$$ where $B^{H_{1}}$ and $B^{H_{2}}$ are two independent fractional Brownian motions, K is a kernel function and the bandwidth parameter α satisfies certain hypotheses in terms of H 1 and H 2. Its limiting distribution is a mixed normal law involving the local time of the fractional Brownian motion $B^{H_{1}}$ . We use the techniques of the Malliavin calculus with respect to the fractional Brownian motion.

Suggested Citation

  • Solesne Bourguin & Ciprian A. Tudor, 2012. "Asymptotic Theory for Fractional Regression Models via Malliavin Calculus," Journal of Theoretical Probability, Springer, vol. 25(2), pages 536-564, June.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:2:d:10.1007_s10959-010-0302-y
    DOI: 10.1007/s10959-010-0302-y
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    References listed on IDEAS

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    1. Phillips, Peter C B, 1988. "Regression Theory for Near-Integrated Time Series," Econometrica, Econometric Society, vol. 56(5), pages 1021-1043, September.
    2. Wang, Qiying & Phillips, Peter C.B., 2009. "Asymptotic Theory For Local Time Density Estimation And Nonparametric Cointegrating Regression," Econometric Theory, Cambridge University Press, vol. 25(3), pages 710-738, June.
    3. Qiying Wang & Peter C. B. Phillips, 2009. "Structural Nonparametric Cointegrating Regression," Econometrica, Econometric Society, vol. 77(6), pages 1901-1948, November.
    4. Coutin, Laure & Nualart, David & Tudor, Ciprian A., 2001. "Tanaka formula for the fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 94(2), pages 301-315, August.
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