Nonparametric Regression with Subfractional Brownian Motion via Malliavin Calculus

Abstract

We study the asymptotic behavior of the sequence Sn=∑i=0n-1K(nαSiH1)(Si+1H2-SiH2), as n tends to infinity, where SH1 and SH2 are two independent subfractional Brownian motions with indices H1 and H2, respectively. K is a kernel function and the bandwidth parameter α satisfies some hypotheses in terms of H1 and H2. Its limiting distribution is a mixed normal law involving the local time of the sub-fractional Brownian motion SH1. We mainly use the techniques of Malliavin calculus with respect to sub-fractional Brownian motion.