Asymptotic theory for fractional regression models via Malliavin calculus

Abstract

We study the asymptotic behavior as $n\to \infty$ of the sequence $$S_n=\sum _{i=0}^{n-1}K(n^{\alpha }{B}_i^{H_1})(B_{i+1}^{H_2}-B_i^{H_2})$$ 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.