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Asymptotic Theory for Fractional Regression Models via Malliavin Calculus

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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.

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Authors and Affiliations

  1. SAMM, Université de Paris 1 Panthéon-Sorbonne, 90, rue de Tolbiac, 75634, Paris, France

    Solesne Bourguin

  2. Laboratoire Paul Painlevé, Université de Lille 1, 59655, Villeneuve d’Ascq, France

    Ciprian A. Tudor

Authors
  1. Solesne Bourguin
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  2. Ciprian A. Tudor
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Correspondence to Ciprian A. Tudor.

Additional information

C.A. Tudor is associate member of the team Samm, Université de Panthéon-Sorbonne Paris 1.

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Bourguin, S., Tudor, C.A. Asymptotic Theory for Fractional Regression Models via Malliavin Calculus. J Theor Probab 25, 536–564 (2012). https://doi.org/10.1007/s10959-010-0302-y

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  • DOI: https://doi.org/10.1007/s10959-010-0302-y

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