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Nonparametric regression with subfractional Brownian motion via Malliavin calculus. (English) Zbl 1474.62119

Summary: We study the asymptotic behavior of the sequence \(S_n = \sum_{i = 0}^{n - 1} K(n^\alpha S_i^{H_1})(S_{i + 1}^{H_2} - S_i^{H_2})\), 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 \(\alpha\) 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.

MSC:

62G08 Nonparametric regression and quantile regression
60H07 Stochastic calculus of variations and the Malliavin calculus
60G22 Fractional processes, including fractional Brownian motion
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