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.
Categories: math.PR