Nonparametric Regression with Subfractional Brownian Motion via Malliavin Calculus

Abstract EN

We study the asymptotic behavior of the sequence S n = ∑ i = 0 n - 1 K ( n α S i H 1 ) ( S i + 1 H 2 - S i H 2 ) , as n tends to infinity, where S H 1 and S H 2 are two independent subfractional Brownian motions with indices H 1 and H 2 , respectively.

K is a kernel function and the bandwidth parameter α satisfies some hypotheses in terms of H 1 and H 2 .

Its limiting distribution is a mixed normal law involving the local time of the sub-fractional Brownian motion S H 1 .

We mainly use the techniques of Malliavin calculus with respect to sub-fractional Brownian motion.