Asymptotic behavior for bi-fractional regression models via Malliavin calculus

Abstract

Let B^H1,K1 and B^H2,K2 be two independent bi-fractional Brownian motions. In this paper, as a natural extension to the fractional regression model, we consider the asymptotic behavior of the sequence

S_n:={\displaystyle \sum _{i=\mathrm{0}}^{n-\mathrm{1}}K(n^{\alpha }{B}_i^{H_{\mathrm{1}},K_{\mathrm{1}}})}(B_{i+\mathrm{1}}^{H_{\mathrm{2}},K_{\mathrm{2}}}-B_i^{H_{\mathrm{2}},K_{\mathrm{2}}}),

where K is a standard Gaussian kernel function and the bandwidth parameter αsatisfies certain hypotheses. We show that its limiting distribution is a mixed normal law involving the local time of the bi-fractional Brownian motion B^H1,K1. We also give the stable convergence of the sequence S_n by using the techniques of the Malliavin calculus.

Keywords

Bi-fractional Brownian motion (bi-fBm) / Malliavin calculus / regression model

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