Nonparametric Regression with Subfractional Brownian Motion via Malliavin Calculus

Abstract

We study the asymptotic behavior of the sequence $\mathrm{}{S}_n={\sum }_{i=\mathrm{0}}^{n-\mathrm{1}}K(n^{\alpha }{S}_i^{H_{\mathrm{1}}})(S_{i+\mathrm{1}}^{H_{\mathrm{2}}}-S_i^{H_{\mathrm{2}}}),$ as $n$ tends to infinity, where $S^{H_{\mathrm{1}}}$ and $S^{H_{\mathrm{2}}}$ are two independent subfractional Brownian motions with indices $H_{\mathrm{1}}$ and $H_{\mathrm{2}}$, respectively. $K$ is a kernel function and the bandwidth parameter $\alpha$ satisfies some hypotheses in terms of $H_{\mathrm{1}}$ and $H_{\mathrm{2}}$. Its limiting distribution is a mixed normal law involving the local time of the sub-fractional Brownian motion $S^{H_{\mathrm{1}}}$. We mainly use the techniques of Malliavin calculus with respect to sub-fractional Brownian motion.