Asymptotic theory for fractional regression models via Malliavin calculus

Details Asymptotic theory for fractional regression models via Malliavin calculus Asymptotic theory for fractional regression models via Malliavin calculus

2010-04-05

arxiv.org/abs/1004.0680v1

Mathematics

Probability

Scientific paper

We study the asymptotic behavior as $n\to \infty$ of the sequence $$S_{n}=\sum_{i=0}^{n-1} K(n^{\alpha} B^{H_{1}}_{i}) (B^{H_{2}}_{i+1}-B^{H_{2}}_{i})$$ where $B^{H_{1}}$ and $B^{H_{2}}$ are two independent fractional Brownian motions, $K$ is a kernel function and the bandwidth parameter $\alpha$ 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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