Asymptotic Theory for Fractional Regression Models via Malliavin Calculus

Solesne Bourguin () and Ciprian A. Tudor ()
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Solesne Bourguin: Université de Paris 1 Panthéon-Sorbonne
Ciprian A. Tudor: Université de Lille 1

Journal of Theoretical Probability, 2012, vol. 25, issue 2, 536-564

Abstract: 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.

Keywords: Limit theorems; Fractional Brownian motion; Multiple stochastic integrals; Malliavin calculus; Regression model; Weak convergence; 60F05; 60H05; 91G70 (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/s10959-010-0302-y

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